NELSON'S 
i  'OMMON-8CHOOL ARITHMETIC: 


THE   LOW 


I  THE  METEJ 


I  if 

In- 


ME  As 


LISH 


IN  MEMORIAM 
111     FLOR1AN  CAJORI 


NELSON'S; 
COMMON-SCHOOL  ARITHMETIC: 

/> 

DESIGNED   FOE,  THE   USE   OF  THE   LOWEST  AS 
WELL   AS   THE   HIGHEST   CLASSES, 

AND    CONTAINING 

THE   APPLICATION   OF  ARITHMETIC   TO   THE 
GENERAL  PURPOSES  OF  LIFE, 

AND 

THE  METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES, 

KECENTLY  ADOPTKT)  BY  CONGRESS. 

Bv   RICHARD   NELSON, 

PRESIDENT  OF  NELSOX'S  VNTON  BlTSTNESS  COLLEGE,  AND  AUTHOR  OP 
NELSON'S  MERCANTILE  ARITHMETIC. 


SECOIT3D     lEZDITIOIbT. 


u 


CINCINNATI: 
R.  W.  CARROLL   &   CO.,    PUBLISHERS, 

No.  117  WEST  FOURTH  STREET. 

18G7. 


\ 


Entered  according  to  Act  of  Congress,  in  the  year  1867,  by 
KIC1IARD  NELSON, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the  Souther* 
District  of  Ohio. 


PREFACE. 


THIS  treatise  differs  so  materially  from  others  of  ita 
class,  that  the  space  of  a  preface  will  not  suffice  to  give 
reasons  for  the  changes  made.  The  author  will,  therefore, 
have  to  content  himself  with  stating  that,  in  the^lassifi^ 
catjoj]  and  treatment  of  subjects,  he  has  been  guided  by 
his  own  experience  and  that  of  the  most  distinguished 
educators  of  the  country  ;  and  in  the  application  of  the 
science  to  the  purposes  of  life,  his  authorities  have  been 
practical  men,  who  were  familiar  with  the  defects  of  the 
old  system,  and  desirous  that  the  rising  generation  should 
be  trained  with  more  direct  reference  to  their  probable 
future  callings. 

improvements  in  machinery  having  almost  super-  ^ 


seded  the  necessity  for  a  knowledge  of  arithmetic  in 
mechanic  arts,  little  has  been  done  to  adapt  the  treatise 
to  such  purpose;  hence,  it  partakes  largely  of  the  mercan- 
tile  character. 


This,  the  author  considers,  will  be  its  highest  recom- 
mendation, as  the  destination  of  most  American  youth  ia 
business;  and,  especially,  as  every  man  in  this  great  Na- 
tion of  Commerce  is  more  or  less  engaged  in  mercantile 
pursuits. 

(iii) 


iy  PREFACE. 

The  author  acknowledges  numerous  acts  of  kindness 
and  courtesy  received,  in  the  preparation  of  this  work, 
from  officers  of  the  Government,  gentlemen  of  the  legal 
and  mercantile  professions,  teachers  in  different  parts  of 
the  country,  especially  from  the  Principals  of  the  Pub- 
lic Schools  of  Cincinnati,  most  of  whom  generously  and 
unreservedly  tendered  advice  regarding  topics  of  interest 
to  their  schools.  To  the  distinguished  educators,  A.  J. 
lliCKOFP  and  JOHN  HANCOCK — the  latter  the  friend  and 
associate  of  the  author — he  is  under  peculiar  obligations. 
The  former  gentleman,  having  been  the  first  to  adopt  his 
Mercantile  Arithmetic,  was  the  most  competent  to  suggest 
further  improvements  and  hints  for  the  adaptation  of  arith^ 
metic  to  common  school  purposes;  while  the  latter,  from 
his  long  experience  as  Principal  of  the  First  Intermediate 
School  in  this  city,  and  his  immediate  knowledge  of  the 
wants  of  teachers,  proved  a  valuable  companion  during  the 
progress  of  the  work,  aiding  constantly  in  the  revision  of 
the  proof-sheets  and  tendering  professional  advice. 

Though  little  reliance  has  been  placed  on  written  au- 
thority, many  works  have  been  called  into  requisition, 
principally  of  a  legal,  scientific  and  educational  character. 
Of  these,  free  use  has  been  made,  especially  in  the  de- 
partment of  the  tables,  which,  it  is  hoped,  will  be  found 
to  subserve  the  present  wants  of  every-day  life. 

liicHARD  NELSON. 
CINCINNATI,  August,  1SOG. 


A  VOCABULARY 

OF 

TECHNICAL  TERMS  USED  IN  BUSINESS. 


ABATEMENT,  or  deduction,  an  amount  taken  off  a  bill  for  prompt  payment, 

damages,  etc. 
ACCEPTANCE,  agreeing  to  price  or  terms  proposed;  a  bill  with  one'8  name 

written  in  such  a  way  as  to  bind  for  payment.    See  page  197. 
ACCOMMODATION  PAPER,  a  bill  or  note  used  to  raise  money,  and  not  to  pay 

a  debt. 

ACCOUNT,  detailed  statement  of  goods  sold.    A  statement  showing  the  indebt- 
edness of  one  person  to  another. 

ACCOUNTANT,  a  professional  calculator  ;  one  skilled  in  book-keeping. 
ACCOUNT-BOOK,  a  ruled  book  in  which  accounts  are  kept, 
ACCOUNT  CURRENT,  a  plain  statement  of  a  running  account  between  two 

persons.    See  page  194. 
ACCOUNT  SALES,  a  detailed  statement  of  goods  sold,  made  by  an  agent  to  his 

principal. 

ACQUITTANCE,  a  written  discharge  ;  a  receipt  in  full  for  money  due. 
AD  VALOREM,  according  to  value,  an  assessment  for  custom  duty. 
ADVANCE,  a  sum  of  money  paid  before  value  is  received. 
ADVENTURE,  a  doubtful  speculation ;  a  term  used  in  book-keeping  for  goods 

shipped  to  be  sold  on  commission. 
ADVICE,  mercantile  intelligence. 

AFFIDAVIT,  a  declaration  in  writing,  made  on  oath  before  a  magistrate,  etc. 
AGENT,  one  who  acts  for  another. 
ANNUITY,  a  sum  of  money  paid  periodically. 
ANNUL,  to  make  void  ;  to  cancel. 
ANTEDATE,  to  date  beforehand. 

APOTHECARIES'  WEIGHT,  the  weight  used  in  compounding  medicines. 
APPRAISER,  a  valuator. 
ARBITRATION,  reference  of  a  controversy  or  dispute  to  persons  chosen  by  the 

parties. 
ASSESSOR,  or  Surveyor,  one  whose  duty  it  is  to  estimate  the  value  of  property 

for  taxation. 

ASSETS,  the  funds  and  property  of  a  trader  or  person  in  business. 
ASSIGNEE,  one  to  whom  an  assignment  is  made. 

5 


G  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

ASSIGNMENT  a  conditional  transfer  of  property,  making  it  over  for  safe  keep- 
ing. 

ASSIGNOR,  one  who  makes  an  assignment. 
ASSURANCE.     See  Insurance. 

AVERAGE,  allowance  made  for  loss  at  sea;  a  rule  in  arithmetic. 
AUDIT,  to  examine  books,  vouchers,  etc. 
AUDITOR,  one  who  inspects  or  examines  and  certifies  accounts. 

B. 

BAGGAGE,  the  wearing  apparel,  trunks,  etc.,  of  a  traveler. 

BALANCE,  a  well-known  instrument  for  weighing ;  to  find  the  difference  be- 
tween two  sides  of  an  account',  also  that  difference. 

BALANCE  OF  TRADE,  the  difference  of  the  money  value  of  the  produce  re- 
ceived  and  exported.  If  a  country  receives  more  produce  and  manufac- 
tures than  it  ships,  the  balance  of  trade  is  said  to  be  against  it. 

BALANCE-SHEET,  a  paper  containing  a  concise  statement  of  a  merchant's 
accounts. 

BALANCING  BOOKS,  the  business  of  making  a  balance-sheet  from  the  ac- 
counts in  the  ledger. 

BALE,  a  package  of  goods  or  produce. 

BANCO,  a  word  used  as  a  prefix  to  paper  money  of  some  parts  of  Europe. 

BANK-BOOK,  the  pass-book  of  a  bank. 

BANKER,  a  dealer  in  money. 

BANK  HOURS,  from   9  to  3  o'clock. 

BANK  NOTE,  a  bank-bill  payable  to  bearer. 

BANKRUPT,  one  who  is  not  able  to  pay  his  debts. 

BANKS.    See  page  154, 

BANK  STOCK,  the  shares  of  a  banking  company. 

BEAR,  a  term  used  to  designate  a  person  who  makes  it  his  business  to  depress 
the  price  of  stocks,  in  order  to  buy  up. 

BILL  OF  ENTRY,  a  list  of  goods  entered  at  the  custom-house. 

BILL  OF  EXCHANGE.    See  page  190. 

BILL  OF  LADING,  a  receipt  from  a  railroad,  ship,  etc.,  for  goods  entered  for 
conveyance  from  one  place  to  another. 

BILL-HEAD,  a  printed  form,  with  name  of  business  or  address. 

BILL,  or  BILL  OF  PARCELS,  a  detailed  account  of  goods  sold. 

BILL  OF  SALE,  a  contract  under  seal  for  the  sale  of  goods. 

BILLS  PAYABLE,  the  name  given  by  a  merchant  or  other  person  to  notes 
made  and  issued,  or  bills,  drafts,  etc.,  accepted  by  him. 

BILLS  RECEIVABLE,  all  notes  taken  or  given  in  payment,  except  one's  own. 

BLANK  CREDIT,  permission  given  by  house  or  person  to  draw  money  on  ac- 
count. 

BON  A  FIDE,  in  good  faith. 

BOND,  a  note  or  deed  given  with  pecuniary  security. 

BONDED  GOODS,  those  for  which  bonds  are  given  for  the  duties  instead  of 
money. 

BROKER,  an  agent  or  factor.    See  Broker,  page  95. 

BROKERAGE,  the  percentage,  commission,  etc.,  paid  to  a  broker  for  buying 
or  selling. 


VOCABULARY  OF  TECHNICAL  TERMS.  7 

BULL,  a  term  applied  to  a  broker  or  stock  jobber,  who  interests  himself  to 
raise  the  price  of  stocks  in  the  market,  in  order  to  command  a  high  sum 
for  those  he  holds. 

BULLION,  uncoined  gold  and  silver. 

C. 

CAPITAL,  stock  in  trade ;  the  amount  of  assets  employed  by  a  person  or  com- 
pany  in  business. 

CAPITALIST,  a  man  of  large  property  or  means ;  one  who  has  large  sums  in- 
vested in  stocks. 

CAPITATION,  a  poll  tax;  a  tax  levied  on  male  adults. 

CARGO,  a  ship's  load. 

CARRIAGE,  the  charge  made  for  conveying  goods  from  one  place  to  another. 

CARTAGE,  the  charge  for  carrying  goods  on  a  cart. 

CASE,  a  box  for  holding  goods  or  merchandise. 

CASH,  the  general  name  for  coin  and  bank  notes  ;  sometimes  checks  and  sight 
bills  of  exchange  are  called  cash. 

CASH-BOOK,  the  book  in  which  merchants  and  others  enter  the  money  paid 
out  and  taken  in. 

CASH  CREDIT,  the  privilege  of  drawing  money  at  a  bank,  obtained  by  depos- 
iting suitable  security. 

CASHIER,  one  who  has  charge  of  money. 

CELLARAGE,  privileged  charge  of  rooms  underground. 

CERTIFICATE,  testimony  given  in  writing;  a  paper  granting  some  particular 
privilege. 

CHAMBER  OF  COMMERCE,  an  association  of  merchants  for  the  protection 
of  trade. 

CHARTER,  license  from  government  to  pursue  certain  kinds  of  business. 

CHARTER  PARTY,  a  contract  in  writing  between  the  owner  and  freighter  of  a 
vessel. 

CHATTELS,  all  goods  and  real  or  personal  property,  except  real  estate. 

CHECK,  an  order  on  a  bank  for  payment  on  demand. 

CHECK-BOOK,  a  printed  book  of  blank  checks. 

CHEST,  a  box  or  package;  tea  and  opium  are  packed  in  chests.  A  chest  of 
opium  contains  111%  Ibs. ;  tare  allowed,  \%  Ibs.  A  chest  of  tea  is  variable. 

CIRCULAR,  a  printed  letter  of  advertisement. 

CLOSING  AN  ACCOUNT,  balancing  the  two  sides,  by  placing  the  difference 
on  the  smaller  side,  under  the  name  of  balance  or  profit  and  loss,  and 
drawing  lines  beneath. 

CLERK,  an  assistant  in  a  store,  office,  etc.    See  page  198. 

COLLECTOR,  one  authorized  to  receive  money  for  another. 

COMMERCE,  the  business  of  exchanging  one  commodity  for  another;  buying 
and  selling;  mercantile  business. 

COMMERCIAL,  pertaining  to  commerce. 

COMMISSION.    See  Commission,  page  139. 

COMPANY,  a  number  of  persons  associated  in  business. 

COMPENSATION,  remuneration  or  reward  for  injury  or  services. 

COMPETITION,  rivalry,  contention  for  contract  or  supremacy. 

CONSIDERATION,  a  bonus;  a  sum  given  on  account  for  any  thing. 

CONSIGN,  to  send  goods  to  an  agent  or  factor  for  sale. 


8  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

CONSIGNEE,  a  person  \vho  receives  goods  on  trust,  or  to  be  sold  on  com* 

mission. 

CONSIGNMENT,  goods  sent  to  a  distance  for  sale  by  an  agent. 
CONSIGNOR,  the  person  or  party  who  consigns. 
CONSOLS,  public  stocks  in  England. 
CONSUMER,  one  who  uses  or  expends  goods. 
CONSUMPTION,  a  using  up;  the  quantity  consumed. 
CONTINGENT,  a  share  arising  from  an  adventure;  doutbful. 
CONTRA,  on  the  other  side  ;  per  contra,  a  writing  on  the  opposite  side. 
CONTRABAND  GOODS,  articles  on  which  there  are  heavy  duties,  or  those 

wholly  prohibited  by  government. 

CONTRACT,  an  agreement  between  parties  for  a  lawful  consideration  ;  a  bargain. 
CONTRACTOR,  one  who  bargains. 

CONTRIBUTION,  a  joint  payment  of  money  to  an  undertaking. 
CONVEYANCE,  a  legal  document,  transferring  land  or  other  property  from 

one  person  to  another;  the  transport  of  goods  or  passengers  from  one  placo 

to  another  ;  a  vehicle. 
COOPERAGE,  money  paid  to  a  cooper,  or  charges  made  for  repairs  of  casks, 

etc. 

COPARTNER  a  person  engaged  in  partnership. 

COPYING  INK,  adhesive  ink,  prepared  with  gum,  etc.,  for  transferring  writing. 
COPYING  PRESS,  an  instrument  for  taking  impressions  from  writing;  copy- 
ing letters,  etc. 

CORRESPONDENT,  a  letter  writer ;  news  writer  for  a  periodical. 
COUNTER  ENTRY,  a  contrary  entry. 
COUNTERMAND,  a  contrary  order. 

COUNTING-HOUSE,  or  COUNTING-ROOM,  a  merchant's  office. 
COUPON,  that  part  of  a  bond  or  other  instrument  designed  to  be  cut  off. 
CREDIT,  giving  trust;  goods  supplied  without  present  payment. 
CURRENCY,  paper  money  and  coin  established  as  the  circulating  medium  of 

a  country. 

CUSTOM,  a  tax  levied  on  goods  imported  or  exported. 
CUSTOMER,  a  regular  buyer  of  goods  at  a  stated  place. 
CUSTOM-HOUSE,  the  place  appointed  to  receive  custom. 
CUSTOM-HOUSE  ENTRY,  a  statement  made,  and  fees  and  expenses  paid  in 

clearing  a  ship. 

D. 

DAMAGE,  injury  inflicted  or  sustained. 

DAMAGED  GOODS,  articles  of  merchandise  or  produce  which  have  been  in- 
jured. 

DAY-BOOK,  the  book  in  which  merchants  record  daily  transactions. 

DAYS  OF  GRACE.    See  page  142. 

DEBIT,  to  make  any  thing  debtor  in  one's  books  ;  a  charge  entered 

DEBIT  SIDE,  the  left  side  of  a  page  in  the  ledger. 

DEBT,  something  due  to  another. 

DEBTOR,  one  who  owes  another. 

DECIMAL  CURRENCY,  moneys  reckoned  by  tens,  as  the  United  States  cur- 
rency. 

DEED,  a  legal  instrument  of  agreement  under  seal. 


VOCABULARY  OF  TECHNICAL  TERMS.  9 

DEFAULT,  a  failure  of  payment. 

DEFAULTER,  one  who  makes  away  with  the  public  funds  intrusted  to  his 
en  re. 

DEFICIT,  a  deficiency;  something  wanted 

DEPOSIT,  a  lodgment;  a  pledge  or  pawn;  money  intrusted  to  the  care  of 
others, 

DEPOSITOR,  one  who  has  money  lodged  in  bank  for  safe  keeping. 

DEPRECIATION,  lessening  in  value. 

DESPATCH;  to  transmit,  to  forward  goods,  papers,  etc.    See  Dispatch. 

DETERIORATION,  damage  done  ;  wear  and  tear  sustained. 

DIRECTOR,  a  manager,  a  superintendent  selected  by  a  company  or  board. 

DIRECTORY,  an  alphabetical  guide  or  address  book  to  the  inhabitants  of  a  city. 

DISCOUNT,  a  deduction  ;  something  thrown  off  the  amount  of  a  bill  or  note; 
the  sum  paid  by  way  of  interest  for  the  advance  of  money  at  bank. 

DISCOUNT  BROKER,  one  who  loans  money  on  notes  of  hand. 

DISCOUNT  DAY,  some  banks  discount  only  on  stated  days,  called  discount 
days. 

DISPATCH,  a  letter  or  message  by  telegraph. 

DISSOLUTION  breaking  up  of  a  copartnership. 

DITTO,  the  same. 

DIVIDEND,  interests  on  stocks ;  a  share  of  the  proceeds  of  a  joint  stock  spec- 
ulation. 

DOCK,  a  secure  landing  for  ships ;  a  place  for  landing  cargoes ;  also  a  place  to 
build  or  repair  ships. 

DOUBLE  ENTRY,  a  method  of  keeping  books,  which  considers  every  busi- 
ness transaction  contains  both  a  debit  and  a  credit. 

DRAFT,  an  order  to  pay  money;  a  deduction  from  the  weight  of  goods;  a 
rough  copy  of  a  writing,  etc. 

ORAW,  to  write  an  order  on  another  for  money  or  goods. 

DRAWEE,  the  person  on  whom  the  bill  is  drawn. 

DRAWER,  the  person  who  draws  a  bill. 

DRAYAGE,  the  charge  made  for  goods  carried  on  a  dray. 

DRUGGIST,  one  who  sells  drugs,  chemicals,  paints,  etc. 

DRY  GOODS,  a  commercial  name  for  cottons,  woolens,  laces,  etc.  In  England, 
for  grain,  coal,  etc. 

DUPLICATE,  a  copy ;  a  second  article  of  a  kind. 

DUTY,  a  tax  on  goods  or  merchandise. 

E. 

EFFECTS,  goods,  property  on  hand,  one's  possessions. 

ENDORSE,  to  employ  to  the  exclusion  of  every  thing  else.    See  Indorse. 

ENTERPRISE,  an  adventure ;  a  projected  scheme. 

ENTRY,  a  record  made  in  a  business  book ;  depositing  a  ship's  papers  ofc. 
landing. 

ENGROSS,  to  monopolize. 

ESTIMATE,  to  appraise  or  value  ;  to  judge  by  inspection. 

EXCHANGE,  giving  one  commodity  for  another;  a  place  of  meeting;  per- 
centage arising  from  the  sale  of  bills,  etc. 

EXECUTOR,  a  person  appointed  to  carry  out  the  intentions  of  a  testator. 

EXHIBIT,  a  voucher  or  document  produced  in  a  court  of  law. 


10  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

EXPENDITURE,  a  charge  or  disbursement;  outlay  for  expenses. 

EXPORTER,  a  shipper  who  sends  goods  or  produce  to  another  country  for  sale. 

EXPORTS,  goods  sent  out  of  a  country. 

EXPRESS,  a  special  messenger;  a  species  of  conveyance. 

F. 

FACE,  the  amount  for  which  a  note  is  drawn ;  also  the  side  on  which  the  writ- 
ing is  made. 

FAC-SIMILE,  an  exact  copy. 

FACTOR,  an  agent  or  broker. 

FAILURE,  a  term  for  suspension  of  payment;  breaking  up  of  business. 

FANCY  GOODS,  ribbons,  silks,  satins,  etc. 

FEE,  a  gratuity  ;  the  charge  of  a  professional  man  for  services. 

FEE  SIMPLE,  a  property  acquired  by  inheritance  or  that  owned  without  con- 
ditions. 

FELLOWSHIP,  companionship,  partnership. 

FINANCE,  ready  money,  funds  or  resources. 

FINANCIER,  one  skilled  in  money  matters. 

FIRE  INSURANCE,  security  against  loss  from  fire,  obtained  by  the  payment 
of  a  small  fee. 

FIRE  POLICY,  the  document  received  from  an  insurance  house  when  goods 
are  insured. 

FIRM,  a  copartnership,  a  house  of  business. 

FLAT,  low,  dull,  inactive. 

FLUSH,  full ;  an  abundance  of  money. 

FORESTALL,  to  buy  up  goods  or  produce  before  the  regular  time  of  sale. 

FOLIO,  page. 

FORWARDER,  an  agent  who  attends  to  the  conveyance  of  goods,  etc. 

FORWARDING  HOUSE,  merchants  who  forward  goods  from  one  place  to  an- 
other. 

FREIGHT,  a  load ;  charge  made  for  carrying  goods  on  ship  or  railroad. 

FUNDS,  ready  money.    See  Public  Funds. 

FURS,  preserved  skins  of  wild  animals,  with  fine  thick  hair. 

FUR  TRADE,  the  business  of  dealing  in  furs. 

G. 

GAUGE,  to  measure  the  contents  of  vessels,  or  barrels,  casks,  etc. ;  a  measure 
or  standard. 

GOODS,  a  general  name  for  movables. 

GROCER,  a  dealer  in  sugar,  spices,  dried  fruits  and  articles  of  food  for  the  table. 

GROSS,  the  whole  weight  of  merchandise  and  box,  barrel,  etc. ;  12  dozen;  a 
great  gross  is  12  times  12  dozen. 

GUARANTEE,  or  WARRANTY,  indemnity  against  loss;  one  who  binds  him 
self  to  see  the  stipulation  of  another  performed. 

GUNNY  BAGS,  coarse  sacking  made  in  India,  used  for  holding  coffee,  rice,  etc. 

H. 

HAND,  a  measure  of  four  inches,  used  for  taking  the  height  of  horses. 

HARDWARE,  goods  manufactured  from  iron. 

HAWKER,  a  peddler. 

HOGSHEAD,  a  large  cask,  formerly  a  measure  of  capacity. 

HORSE  REPOSITORY,  a  place  kept  for  the  sale  of  horses. 


VOCABULARY  OF  TECHNICAL  TERMS.       11 

HUNDRED  WEIGHT,  a  hundred  pounds.    In  England,  112  pounds 
HONOR,  to  accept  a  draft,  by  paying  or  promising  to  pay. 
HYPOTHECATE,  to  pledge  as  security. 

I. 

IMMOVABLES,  lands,  houses,  fixtures,  etc. 

IMMUNITY,  freedom  from  tax,  office  or  obligation. 

IMPERATIVE,  positive,  commanding. 

IMPERISHABLE,  not  subject  to  decay  or  waste. 

IMPORTED,  brought  from  a  foreign  country. 

IMPORTER,  one  who  brings  goods  from  abroad. 

INCOME,  receipts,  gains  from  labor,  trade,  etc. 

INCONVERTIBLE,  not  transmissible ;  ftmds  that  can  not  be  converted 

stocks. 

INDORSE,  to  write  one's  name  on  the  back  of  a  note  or  draft. 
INDORSEMENT,  a  writing  on  the  back  of  a  note  or  other  paper 
INDORSER,  one  who  makes  an  indorsement. 
INITIALS,  the  first  or  capital  letters  of  a  name. 
INLAND  BILLS.    See  page  196. 
INTEREST,  right  or  share  in  business.    See  page  142. 
INSOLVENT,  want  of  ability  to  pay. 
INSURANCE.    See  page  244. 

INTELLIGENCE-OFFICE,  a  registry  office  for  domestics  looking  for  situa- 
tions. 

INVENTORY,  a  list  of  goods  or  effects. 
INVESTMENT,  capital  employed  ;  money  at  interest. 
INVOICE.    See  page  100. 
INVOICE  BOOK,  a  book  containing  invoices  or  copies  of  invoices. 

J. 

JOINT-STOCK  COMPANY,  an  association  of  men  to  carry  on  heavy  under- 
takings. 
JOURNAL,  an  intermediate  book  between  a  day-book  and  a  ledger. 

L. 

LAND-WARRANT,  a  title  to  a  lot  of  public  land. 

LEASE,  a  deed  ;  a  contract  for  the  use  of  property. 

LEGAL  TENDER,  the  authorized  coins  or  money  of  a  country. 

LETTER  OF  ADVICE,  intelligence. 

LETTER  OF  ATTORNEY,  legal  authority  to  act  for  another. 

LETTER  OF  CREDIT,  a  letter  from  a  mercantile  or  banking-house  given  to  a 

traveler,  by  which  he  can  collect  money  in  a  foreign  country. 
LEVEE,  shipping  place  or  landing. 
LICENSE,  a  grant. 

LIEN,  a  legal  claim  ;  power  to  prevent  sale  by  another. 
LIGHTERAGE,  charges  for  conveying  goods  by  a  lighter. 
LIQUIDATION,  the  act  of  settling  debts. 

LIVE  STOCK,  animals  kept  on  a  farm  or  for  sale,  as  cows,  horses,  hogs,  etc. 
LLOYDS,  an  establishment  in  London  for  the  classification  of  ships;  a  place 

of  assembly  for  merchants  and  underwriters  to  assemble. 
LUGGAGE,  baggage,  the  clothing,  etc.,  of  a  traveler. 


12  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

M. 

MANIFEST,  a  list  or  exhibit  of  a  vessel's  cargo. 
MARINE,  belonging  or  pertaining  to  the  sea. 
MARKET,  a  place  of  sale  ;  price. 
MARKETABLE,  what  may  be  readily  sold. 
MART,  a  market. 

MATERIALS,  the  substances  from  which  goods  and  wares  are  made  up. 
MATURITY,  the  time  when  a  bill  or  note  falls  due. 

MEASUREMENT' GOODS,  light  goods  taken  on  freight  by  measurement. 
MERCHANDISE,  trade  goods  or  wares  ;  goods  bought  to  sell. 
MINT,  an  official  place  for  coining  money. 
MONEY  BROKER,  a  dealer  in  money,  bills  of  exchange,  etc. 
MORTGAGE,  a  pledge  of  land  for  the  payment  of  a  debt. 
MORTGAGEE,  the  person  who  holds  the  pledge. 
MORTGAGER,  the  person  who  gives  the  pledge. 
MOVABLES,  things  that  can  be  moved  easily,  as  furniture,  etc. 

N. 

NEST,  a  set  of  tubs,  buckets,  baskets,  etc. 

NET,  the  clear  amount,  the  quantity  remaining  after  all  deductions. 
NET  PROCEEDS,  the  remainder  after  deducting  expenses. 
NOTARIAL  SEAL,  the  seal  of  a  notary  public. 
NOTARY  PUBLIC,  an  officer  authorized  to  attest  documents  and  protest  bills 

of  exchange,  notes,  etc.,  for  non-payment  or  non-acceptance. 
NOTE,  a  written  promise  to  pay  a  debt ;  a  memorandum. 

O. 

OBLIGATION,  a  bond,  a  binding  agreement. 
ORDER,  a  request  to  pay  ;  commission  given  to  supply  goods. 
ORDER-BOOK,  a  manufacturer's  book,  in  which  orders  are  copied. 

P. 

PACKAGE,  a  bundle. 

PACKER,  a  person  who  receives  goods  to  pack  for  shipment. 
PANIC,  a  monetary  pressure  or  crisis. 
PAPER,  an  article  in  common  use;  the  name  given  by  merchants  to  notes, 

bills,  etc. 

PAPER  CURRENCY,  paper  money  of  a  country. 
PARCEL,  a  small  package  or  bundle. 
PARTNER,  an  associate  ;  the  member  of  a  copartnership. 
PAR  OF  EXCHANGE,  the  value  of  money,  both  in  weight  and  fineness,  when 

compared  with  that  of  other  countries. 
PASS-BOOK,  a  small  book  kept  between  a  bank  and  its  depositors,  a  merchant 

and  his  customers. 

PAYEE,  the  person  to  whom  money  is  to  be  paid. 
PEDDLE,  to  carry  about  goods  for  sale. 

PERSONAL  PROPERTY,  money  and  movable  goods  outside  of  one's  business. 
PETTY  CASH-BOOK,  a  memorandum  book  of  small  receipts  and  expenses. 
POLICY,  a  writing  of  agreement  given  by  insurance  companies. 
POST-DATE,  to  date  after  the  real  time. 
POSTING,  transferring  from  day-book,  journal,  etc.,  to  the  ledger. 


VOCABULARY  OF  TECHNICAL  TERMS.       13 

POWER  OF  ATTORNEY,  authority  to  act  for  another. 

PRICE  CURRENT,  a  published  list  of  market  prices. 

PRIME,  superior. 

PRINCIPAL,  the  hear!  of  a  school  or  business. 

PRO-FORMA,  according  to  form. 

PROMISSORY  NOTE,  an  engagement  in  writing  to  pay  a  specified  sum  at  a 

a  stated  time. 

PROSPECTUS,  outline  or  sketch  of  an  institution,  business,  book,  etc. 
PROTEST,  an  official  notice  from  a  notary  public  of  the  non-payment  of  a  bill, 

preparatory  to  legal  proceedings. 
PURVEYOR,  one  who  supplies  provisions. 

Q- 

QUARTER,  the  fourth  part  of  any  thing;  a  measure  of  weight,  25  Ibs. ;  also  a 
measure  of  length,  9  inches. 

QUOTATIONS,  current  price  for  stock  and  shares,  or  articles  of  produce  in  the 
market. 

R. 

REBATE,  discount,  a  reduction. 

RECEIPT,  an  acquittance,  acknowledgement  of  payment. 

RECEIVER,  a  cashier,  a  person  appointed  to  take  charge  of  property  in  litiga- 
tion. 

RECEIVING  HOUSE,  a  depot  or  store. 

RESOURCES,  funds,  assets,  that  which  may  be  converted  into  supplies. 

RETURNS,  profits  or  receipts  in  business  ;  accounts  of  goods  sold  by  an  agent. 

REMITTANCE,  bills  or  money  sent  from  one  house  to  another. 

RENEWAL  of  a  bill  or  note,  giving  a  new  note  for  a  longer  time  ;  extension  of 
time  on  notes,  etc. 

S. 

SALE,  an  auction  ;  the  disposal  of  goods  to  a  private  bidder. 

SALVAGE,  a  reward  claimed  for  saving  property  from  loss  at  sea. 

SAVINGS  BANKS,  banks  of  deposit,  where  interest  is  allowed  on  the  amount 
lodged 

SCHEDULE,  an  inventory  of  goods  on  parchment  or  paper. 

SCRIP,  a  receipt  or  acknowledgment  for  installments  paid  on  stocks;  a  partial 
receipt,  to  be  substituted  by  a  receipt  in  full  when  all  has  been  paid. 

SECRETARY,  a  head  clerk  or  writer ;  the  recording  officer  of  a  society. 

SHIP-LETTER,  a  letter  forwarded  by  private  ship,  instead  of  a  packet  char- 
tered for  that  purpose. 

SHIPPED,  transmitted  by  sea;  goods  forwarded  by  any  conveyance. 

SHIPPING  CLERK,  a  person  who  attends  to  shipping  of  goods. 

SHIPMENT,  the  goods  forwarded  by  railroad  or  steamboat;  a  term  in  double- 
entry  book-keeping. 

SHOP,  a  work-room  ;  the  name  given  to  stores  in  England. 

SIGHT,  or  AT  SIGHT,  the  time  when  a  bill  is  presented  to  a  person  on  whom 
it  is  drawn. 

SIGNATURE,  the  name  of  a  person  written  by  himself. 

SILENT  PARTNER,  a  partner  who  puts  in  capital,  but  does  not  take  an  active 
pjirt  in  the  business. 

SLEEPING  PARTNER,  the  term  used  in  Britain  for  silent  partner. 


14  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

SMUGGLING,  passing  goods  into  a  country  clandestinely,  so  as  to  avoid  the 

duties. 
STAPLE,  the  commodities  which  always  meet  with  ready  sale;  the  principal 

articles  of  produce  or  manufacture  of  a  country. 
STERLING,  according  to  a  fixed  standard ;  a  term  applied  to  the  money  of 

Great  Britain. 

STOCK,  goods  kept  for  sale  ;  materials  of  manufacture ;  animals  on  a  farm. 
STORAGE,  charge  for  the  use  of  a  warehouse. 
STORES,  supplies  laid  in  for  a  ship. 
SUNDRIES,  in  hook-keeping,  more  than  one;  plurality. 
SUSPENDED,  temporarily  removed  from  employment;   alleged  inability  to 

pay  debts  ;  stoppage  of  work  or  business. 

T. 

TELLER,  an  officer  in  a  bank  who  receives  or  pays  money. 
TIERCE,  a  cask  containing  about  42  gallons. 

TRADE,  the  commerce  of  a  country  ;  to  exchange  commodities;  a  bargain. 
TRADE  ALLOWANCE,  trade  price;  a  discount  allowed  to  merchants. 
TRADESMAN,  a  mechanic;  in  England,  a  storekeeper  or  retailer. 
TRAFFIC,  trade,  exchanging  commodities. 
TRANSCRIPT,  a  copy. 

TRANSFER,  a  change  of  property,  government  funds,  etc. 
TRANSHIPMENT,  the  act  of  removing  from  one  ship  to  another. 
TRANSPORTATION,  the  conveyance  of  goods  ;  a  name  for  a  forwarding  com- 
pany. 
TRANSITU,  on  passage ;  on  the  way  from  one  place  to  another. 

V. 

VENDOR,  a  seller;  one  who  disposes  of  goods  or  property. 
VENDUE,  a  public  sale  ;  an  auction. 

VOUCHER,  an  instrument  of  writing ;  a  document  produced  to  substantiate 
a  statement  of  disbursements. 

W. 
WALL  STREET,  the  street  in  New  York  City  where  tne  principal  bankers  are 

located. 

WAREHOUSE,  store-room  ;  a  place  for  depositing  goods. 
WRIT,  an  official  notice  from  a  law  court. 

To  TEACHERS?.— Attention  is  requested  to  the  arrangement  of  the  answers,  which 
makes  the  book  subserve  the  purpose  of  one  without  answers  and  a  key. 

In  some  places,  as  on  page  44,  the  sum  of  the  answers  to  a  group  of  questions  is 
given.  Should  the  pupil  construe  his  work  to  suit  this  answer,  the  teacher  can 
detect  the  fraud  by  working  auy  one  of  the  questions  of  the  group. 

In  other  places,  as  on  page  40,  the  answers  are  arranged  promiscuously,  with 
usually  one  or  more  than  the  number  of  questions.  This  prevents  copying  or 
working  from  the  answer,  and  yet  encourages  the  learner  to  study. 


MONEY,  WEIGHTS  AND  MEASURES. 


FEDERAL  MONEY  consists  of  four  kinds :  gold,  silver, 
nickel  and  paper.  The  smallest  gold  coin  is  of  the  de- 
nomination of  one  dollar.  Other  gold  coins  are  the  quar- 
ter-eagle, half-eagle,  eagle  and  double-eagle. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dol- 
lar, dime,  half-dime  and  three-cent  piece. 

These  are  also  represented  by  paper  of  the  same  de- 
nomination. 

The  nickel  coins  are  the  one  and  two-cent  pieces. 

The  established  currency  of  the  United  States  consists 
of  the  eagle,  dollar,  dime  and  mill;  but  accounts  are  kept 
in  dollars  and  cents  only. 

TROY  WEIGHT  is  used  in  the  sale  of  gold  and  silver 
and  at  the  mint  for  coinage:  24  grains— 1  pennyweight; 
20  pennyweights=l  ounce ;  12  ounces=l  pound.  The 
signs  are  gr.,  pwt.,  oz. 

The  caret,  when  applied  to  gold,  is  only  a  comparative 
weight,  used  to  indicate  the  proportions  of  pure  gold  and 
alloy.  It  is  ^¥  part  of  the  mass  of  whatever  weight.  18 
carets  fine  is  J|  gold,  or  18  parts  gold  and  6  alloy. 

COMMERCIAL  WEIGHT,  used  in  selling  groceries,  drugs, 
etc.:  16  ounces=l  pound;  2000  pounds— 1  ton.  Signs, 

OZ.,    Us.,   T. 

NOTE. — The  ounce  and  pound  are  the  principal  parts  of  avoir- 
dupois weight  in  use  in  the  United  States.  Iron  ore  and  hemp  are 

(15) 


16  KELSON'S  COMMON-SCHOOL  ARITHMETIC. 

weighed  by  the  old  standard,  112  pounds  to  the  hundred  (civt.)  and 
20  hundreds,  or  2240  pounds,  to  the  ton.  See  Weight  of  a  Toil 
page  17. 

MEASURES  OF  CAPACITY. — The  units  of  measurement 
are  the  gallon  for  liquid,  and  the  bushel  for  dry  measure. 
The  gallon  contains  58872.2  grains  Troy  of  the  standard 
pound  of  distilled  water  at  39°  F.,  weighed  in  air  of  the 
temperature  of  62°,  and  barometer  pressure  30  inches.  It 
contains  nearly  231  cubic  inches. 

The  bushel  contains  543391.89  grains  Troy  of  distilled 
water,  under  the  above  conditions,  and  is  thus  the  Win- 
chester bushel  of  2150.42  cubic  inches. 

DRY  MEASURE,  used  for  measuring  grain,  fruit,  etc. :  2 
pints— 1  quart;  8  quarts— 1  peck;  4  pecks— 1  bushel. 
Signs,  pt.,  qt.j  ph.,  bu. 

NOTES. — 1.  The  U.  S.  bushel  is  a  cylindrical  vessel,  8  inches  deep 
and  18£  diameter,  inside,  and  contains  2150.42  cubic  inches. 

2.  By  statute  in  Ohio,  the  bushel  for  stone  coal,  coke  and  un- 
slacked  lime  contains  2688  cubic  inches,  and  the  measure  should  be 
24  inches  at  the  top,  20  inches  at  the  bottom  and  14.1  deep,  and 
contain  two  bushels. 

3.  The  bushel  of  New  York  State  contains  80  pounds  of  pure 
water,  or  2211.84  cubic  inches. 

LIQUID  MEASURE,*  for  measuring  all  liquids:  4  gills= 
1  pint;  2  pints— 1  quart;  4  quarts— 1  gallon. 

*1.  Liquid  measure  is  the  old  wine  measure,  and  has  superseded 
that  of  beer  and  ale  measure,  both  in  the  United  States  and  Great 
Britain. 

2.  The  gill  is  seldom  used,  while  barrels,  tierces,  etc.,  are  gauged 
and  reckoned  by?gallons. 

3.  The  gallon  contains  231  cubic  inches. 

4.  A  pint  of  water  weighs  1  pound. 

5.  The  capacity  of  cisterns,  vats,  etc.,  is  usually  reckoned  in  bar- 
rels and  hogsheads.     31 J  gallons=l  barrel;  2  barrels,  or  G3  gal- 
lons=l  hogshead. 


MONEY,  WEIGHTS  AND  MEASURES. 


17 


WEIGHTS  OF  PRODUCE  PER  BUSHEL,  according  to  usage 
in  Cincinnati,  and  as  fixed  by  statute  in  Ohio: 


u 
Apples  dried  

?age.  Stat. 
Ibs.   Ibs. 
25      25 

48    48 

34 
60    60 
20 
25 
30 
62    50 
80 
70 
30 
32 
46 
56     56 
70     70 
8 
16 
60     60 
51 
34 
50 
40 
33     32 
56 
25 

Ibs. 
..  2240 

i 
Peaches  dried 

Qsage.  Stat. 
Ibs.    Ib,'. 

33     33 
60     60 
24 
118 
22 
60     60 
50 
56     56 

40 
60 
62     60 
45     45 
66     56 
44     44 
14 
50     50 
14 
50     50 
00 
45 
40 
30 
60 
60    60 
77.6274 

Ibs. 

..  2240 

Barley  ,  

Peas  

Barley  malt,  weight  of 
b;iirs  included  

green  

Plaster  and  hair 

Beans  .  .     ...        ... 

Peanuts   roasted 

Bran  

Potatoes  Irish  

Bran  shorts              .. 

sweet 

Broom-corn  

Rye... 

Buckwheat  

Rye  malt,  wt.  of  bags 
included       .  .    . 

Coal    bituminous 

cnnnel  

Salt  

Charcoal  

Seed,  clover  

Coke  

timothy   

Castor  beans  

flax  

Corn,  shelled  

hemp  

in  ear....  68  arid 
Hair,  plasteriuf  

orchard  grass.... 
Hungarian  grass 
blue  o*rass  

wet  

Hominy  

millet  

I-iime   slacked  

Malt  

sorghurn  

Ship  stuff  

Middlings        •• 

Shorts  ... 

Oats         

Turnips  

(  )nions 

Wheat  

Onion  sets    

Water,  distilled  

WEIGHTS  PER  TON 

Pier  Iron   chill  mold.  . 

Iron  ore.......  

Pig  Iron,  sand  molds... 
Blooms    ...  . 

..  2268 
'>464 

Hemp  

..  2240 

HAV  ... 

.  2000 

THE  WEIGHT  OF  A 

Flour  

PINT  OF 

ounces. 
.  ...    14        Crnsli    snorfir  

ounces. 
17 

Meal          

.     18 

Brown  sugar  

....  18 

Butter.., 

..  15 

Loaf  suecar..., 

.   19 

18  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

WEIGHT  OF  A  CUBIC  FOOT  OF 

Ibs.       I  Ibs. 


Cast  iron 450.55 

Wrought  iron 486.65 

Steel 4S9.8 

Copper 565 

Lead 708.75 

Brass 537.75 

Tin 456 

White  pine 29.56 

Loose  earth  or  sand 95 

Common  soil 1-4 

Strong  soil 127 

Clay 135 

Coal 45  to    55 

Charcoal..... 18  to     18.5 


Yellow  pine 33.81 

White  oak 35.2 

Live  oak 70 

Salt  water  (sea) 64.3 

Freshwater 62.5 

Air 07529 

Steam 03689 

Clay 135 

Sand  113 

Cork 15 

Tallow 59 

Brick 119 

Coke  32 

Ice 58 


23  cubic  feet  of  sand,  18  of  earth  or  17  of  clay  make  a  ton. 

APOTHECARIES'  FLUID  MEASURE,  used  in  compounding 
medicines  :  60  minims— 1  fluid  drachm ;  8  fluid  drachms= 
1  fluid  ounce;  16  fluid  ounces=l  pint;  8  pints=l  gal- 
lon. Signs,  M.j  minim;  /£.,  fluid  drachm;  /,§.,  fluid 
ounce;  Q,  pint;  Cong.,  gallon. 

MEASURES  OF  TIME. — Time  is  divided  into  seconds, 
minutes,  hours,  days,  weeks,  months,  years  and  centuries. 

60  seconds— 1  minute;  60  minutes— 1  hour;  24  hours— 
1  day;  7  days=l  week;  4  weeks— 1  lunar  month;  12 
calendar  months=l  year;  365  days=l  common  year;  366 
days— 1  leap-year ;  365  days  5  hours  48  minutes  49.7  sec- 
onds, or  365|— 1  solar  year. 

A  leap-year  is  exactly  divisible  by  4,  and  has  29  days 
in  February.  1860  and  1864  were  leap-years. 

The  calendar  months  are 

1.  January,    31  days.  7.  July,  31  days. 

2.  February,  28     "  8.  August,        31     " 

3.  March,        31     "  9.  September,  30     " 

4.  April,         30     «  10.  October,       31     « 

5.  May,  31     "  11.  November,  30     " 
ti.  June,          30     "                  12.  December.    31 


MONEY,  WEIGHTS  AND  MEASURES.  19 

Commencing  with  January,  every  other  month  has  31  days  to 
July,  inclusive;  and  commencing  with  August,  every  other  month 
has  31  days  to  December,  inclusive. 

CIRCULAR  MEASURE  is   divided    into   seconds,  minutes 
and  degrees.     60  seconds— 1  minute;  60 
minutes— 1  degree;  360  degrees— 1  cir- 
cumference. 

Signs,  ",  seconds ;    ',  minutes ;    °,  de- 
grees. 

25°  31'  27",  25  deg.,  31  min.  27  sec. 

LINEAR  MEASURE. — Long  Measure  is  used  for  measur- 
ing length,  breadth,  depth  or  distance.  12  inches— 1  foot; 
3  feet— 1  yard ;  5^  yards— 1  rod,  perch  or  pole ;  40  rods= 
1  furlong;  8  furlongs,  or  320  rods— 1  mile.  Signs,  in.> 
inches;/!?.,  feet;  yd.,  yard;  ?*J.,  rod;  fur.,  furlong;  mi.y 
mile. 

1.  In  a  mile  there  are  63360  inches,  5280  feet,  1760  yards,  320  rods. 

2.  The  furlong  is  seldom  used. 

3.  The  12th  part  of  an  inch  is  called  a  line. 

4.  Cloth  is  measured  by  the  yard  and  fractional  parts  of  a  yard. 

MARINE  MEASURE,  for  measuring  distances  at  sea:  6 
feet^l  fathom;  120  fathoms=l  cable  length:  880  fath- 
oms—I mile,  called  a  nautical  or  geographical  mile ;  60 
geographical  or  69.77  statute  miles— 1  degree.* 

The  speed  of  a  ship  at  sea  is  measured  by  an  instru- 
ment called  a  log-line,  the  knots  in  which  correspond  to 
the  number  of  miles  sailed  per  hour.  Knot  is  therefore 
synonymous  with  mile.  A  ship  sailing  at  7  knots  is  mov- 
ing at  the  rate  of  7  miles  an  hour. 

SQUARE  OR  SURFACE  MEASURE. — Surfaces  are  meas- 
ured by  taking  the  length  and  breadth,  by  long  measure, 
and  multiplying  them  together.  The  length  and  breadth 

*The  depth  of  the  sea  is  measured  by  fathoms. 


20  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

of  the  surface  of  a  foot  are  12  inches  each;  hence,  12 
times  12—144  square  inches— 1  square  foot;  9  square 
feet=l  square  yard. 

Land  Measure :  30 J  square  yards=l  square  rod ;  40 
square  rods=l  rood ;  4  roods,  or  10  square  chains— 1 
acre;  640  acres=l  square  mile;  36  square  miles=l  town- 
ship. Signs,  sq.  yds.)  sq.  rds.,  R.,  A. 

1.  Feet  and  even  inches  may  also  be  used  in  measuring  land. 

2.  A  square  of  flooring  or  roofing  is  100  square  feet. 

3.  A  square  mile  is  sometimes  called  a  section. 

4.  A  square  rood  contains  272  j  square  feet;  an  acre,  48560  square 
feet. 

CUBIC  OR  SOLID  MEASURE  includes  three  dimensions, 
length,  breadth  (or  width)  and  thickness  (or  depth)  mul- 
tiplied together.  1728  cubic  inches— 1  cube  foot;  27 
cubic  feet— 1  cubic  yard. 

STONE  MEASURE  is  applied  to  masonry,  which  is  some- 
times paid  for  by  the  foot,  but  usually  by  the  perch. 
24f  or  25  cubic  feet=l  perch ;  the  former  for  private, 
the  latter  for  public  contracts,  as  railroad  or  government 
work. 

Wood  Measure:  Wood  is  sold  by  the  cord,  which  should 
measure  128  cubic  feet  closely  piled,  or  138  feet  if  stowed 
in  a  boat  or  barge.  A  pile  of  wood  8  feet  long,  4  feet 
wide  and  4  feet  thick  contains  a  cord. 

BRICKLAYERS'  MEASURE. — The  common  dimensions  of 
a  brick  are  8  inches  long,  4  inches  broad  and  2  inches 
thick.  There  are  21  bricks  in  a  cubic  foot  of  wall,  in- 
cluding mortar. 

A  wall  8  inches  or  1  brick  in  thickness  contains  14  bricks  to  the 
square  foot  of  surface. 

A  wall  12  inches  or  1 J  bricks  in  thickness  contains  21  bricks  to 
the  square  foot  of  surface. 

A  wall  16  inches  or  2  bricks  in  thickness  contains  28  bricks  to 
the  square  foot  of  surface. 


MONEY,  WEIGHTS  AND  MEASURES.  21 

A  fall  of  T*0  of  an  inch  in  a  mile  will  produce  a  current 
in  rivers. 

Ice  2  inches  thick  will  bear  infantry ;  4  inches,  cavalry 
or  light  guns;  6  inches,  heavy  field  pieces. 

PAPER. — For  Printers.  Sizes  of  paper  made  by  ma- 
chinery : 

Double  imperial,  32  by  44.  Super  royal,  21  by  27. 

Double  super  royal,  27  by  42.  Royal,  19  by  24,  20  by  25. 
Double  medium,  23  by  26,  24  by  Medium,  18J  by  23J. 

37£  and  25  by  38.  Demy,  17  by  22. 

Royal  and  half,  25  by  29.  Folio  post,  16  by  21. 

Imperial  and  half,  26  by  32.  Foolscap,  14  by  17. 

Imperial,  22  by  32.  Crown,  15  by  20. 

A  sheet  folded  in  2  leaves  is  called  a  folio;  in  4  leaves, 
a  quarto;  in  8  leaves,  an  octavo,  or  Svo.;  in  12  leaves,  a  duo- 
decimo, or  12mo.;  in  18  leaves,  an  18mo. ;  in  24  leaves,  a 
24mo. 

Stationers. — 24  sheets— 1  quire;  20  quires— 1  ream. 

Bookbinders  count  from  16  to  20  sheets  to  the  quire  in 
binding  account  books. 

Wrapping  Papers  are  sold  by  the  ream  and  bundle; 
some  reams  are  short  count;  the  long  count  reams  contain 
full  quires. 

SUNDRIES  : 

1  barrel  of  flour=196  Ibs.  3  inches=l  palm. 

1  barrel  of  pork,  etc. =200  Ibs.  4  inches=l  hand. 

1  firkin  of  butter=55  Ibs.  9  inches=l  span. 

12  articles=l  dozen.  3.28  feet=l  meter. 

12  dozen=l  gross.  3.937   or    47  J  inches=l 

144  dozen==l  great  gross.  aune. 

20  articles=l  score. 

GAS. — 1.43  cubic  feet  of  gas  per  hour  give  a  light  equal 
that  of  a  candle;  1.96  feet  equal  4  candles;  3  cubic  feet 
equal  10  candles. 


22  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

HORSE  POWER,  in  machinery,  is  reckoned  at  33000  Ibs. 
raised  1  foot  in  a  minute;  but  the  ordinary  work  of  a 
horse  is  22500  Ibs.  per  minute  for  8  hours. 

STRENGTH  OF  A  MAN. — The  mean  effect  of  the  power 
of  a  man,  unaided  by  a  machine,  is  the  raising  70  Ibs.  1 
foot  high  in  a  second  for  10  hours  a  day— £  of  the  power 
of  the  horse. 

NOTE. — Two  men  working  at  a  windlass,  at,  right  angles  to  each 
other,  can  raise  70  Ibs.  more  easily  than  one  man  can  30  Ibs. 

A  foot  soldier  travels  70  yards,  making  90  steps,  in  one 
minute,  common  time. 

In  quick  time,  86  yards,  making  110  steps. 

In  double-quick,  109  yards,  making  140  steps. 

Average  weight  of  men,  150  Ibs.  each. 

Five  men  can  stand  in  a  space  of  1  square  yard. 

A  man  without  a  load  travels  on  a  level  ground  8J 
hours  a  day,  at  the  rate  of  3.7  miles  an  hour,  or  31 J-  miles 
a  day.  He  can  carry  111  Ibs.  11  miles  in  a  day. 

A  porter,  going  short  distances  and  returning  unloaded, 
can  carry  135  Ibs.  7  miles  a  day.  He  can  carry  in  a 
wheelbarrow  150  Ibs.  10  miles  a  day. 

HAY. — 10  cubic  yards  of  meadow  hay  weigh  a  ton. 
When  the  hay  is  taken  out  of  old,  or  the  lower  part  of 
large  stacks,  8  to  9  cubic  yards  will  make  a  ton. 

HILLS  IN  AN  ACRE. — 3  feet  apart,  there  are  4840  hills 
in  an  acre. 

BRITISH  MONEY,  WEIGHTS  AND  MEASURES. 

In  Great  Britain,  accounts  are  kept  in  pounds,  shillings, 
pence  and  farthings.  4  farthings— 1  penny;  12  pence— 1 
shilling;  20  shillings=l  pound.  Signs,  (farthings  are 
written  as  fractions  of  a  penny,)  d.>  pence;  s.,  shillings; 
£,  pounds.  ^ 


MONEY,  WEIGHTS  AND  MEASURES.  23 

The  coins  are  the  copper  half-penny  and  penny;  silver , 
three-penny,  four-penny,  six-penny,  shilling,  half-crown 
and  crown  pieces;  gold,  the  half-sovereign,  sovereign  and 
guinea.  The  value  of  the  crown  is  5  shillings;  the  sov- 
ereign, 20  shillings;  the  guinea,  21  shillings. 

THE  COMMERCIAL  WEIGHT  is  the  avoirdupois,  of  which 
there  are  in  use  the  ounce,  pound,  stone,  quarter,  hundred 
and  ton.  16  draclims=l  ounce;  16  ounces=l  pound;  14 
pounds— 1  stone;  28  pounds=l  quarter;  4  quarters—-! 
hundred;  20  hundred— 1  ton.  Signs  cZr.,  drachms;  oz., 
ounces;  Z6s.,  pounds;  qrs.,  quarters;  cwt.,  hundreds;  T., 
tons. 

IMPERIAL  MEASURES  OF  CAPACITY  for  all  liquids  and 
dry  goods,  such  as  grain,  potatoes,  etc. : 

1  gill  (gl.)=8.6648  cubic  inches=5  oz.  water. 

4  gillsr^l  pint— 34,65925  cubic  inches=lj  Ibs.  water. 

2  pints  (pts.)— 1  quart=G9.3185  cubic  inches=2£  Ibs  water. 
4  quarts  (qts.)=l  gallon— 277.274  cubic  inches=10  Ibs  water. 
2  gallons  (gal.)— 1  peck— 554.548  cubic  inches=20  Ibs.  water. 
4  pecks  (pk.)=l  bushel=r2218.192  cubic  inches=80  Ibs.  water. 
8  bushels  (bu.)=l  quarterzirl7745.536  cubic  in.=640  Ibs.  water. 
4  quarters  (qr.)=l  chaldron. 

10  quarters=l  last. 

The  largest  measure  for  liquids  is  the  gallon;  the  smallest  for 
grain,  etc.,  the  peck. 

In  London,  a  chaldron  of  coal  contains  36  bushels. 

NOTE. — By  act  of  Parliament,  in  1824,  wine,  ale  and  dry  meas- 
ures were  superseded  by  the  imperial  measures  of  capacity. 

Time  is  divided  into  quarterly  terms  not  recognized  in 
the  United  States. 

In  England. 

Lady  day,  or  1st  term Mar.  25 

Midsummer,  or  2d  term. ..June  24 
Michaelmas,  or  3d  term. ..Sept.  29 
Christmas,  or  4th  term  ...Dec.  25 


In  Scotland. 


Candlemas,  or  1st  term. ...Feb.    2 
Whitsunday,  or  2d  term. ..May  15 

Lammas,  or  3d  term Aug.    1 

Martinmas,  or  4th  term.. .Nov.  11 


Linear,  square  or  superficial,  cubic  measures,   etc.,  are 
the  same  in  both  countries. 


24 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


Articles  sold  by  "tale,"  or  count:  12  articles=l  dozen; 
12  dozen=l  gross;  5  score=:l  hundred;  6  score— 1  long 
hundred. 

MULTIPLICATION  TABLE. 


IX   1-     l 

2X    1-     2 

3X    1-      3 

4X    1=       4 

1X2-     2 

2X    2=      4 

3X    2=      6 

4X    2^      8 

IX    3=      3 

2X    3=      6 

3X    3^      9 

4X    3=    12 

IX    4=     4 

2X    4  =      8 

3X    4=    12 

4X    4r=    16 

IX    5-     6 

2X    5  —    10 

3X    5=    I5 

4X    «r>=    20 

IX    6^      6 

2X    6^    12 

3X    6=    18 

4X    6^    24 

ix  T:=   7 

2X    7  =    14 

3X    7_^    21 

4X    7=r    28 

IX      8  =         8 

2X    8=    16 

3  X    8  =    24 

4X    8=    32 

IX    9-      9 

2X    9=    18 

3X    9  =    27 

4  X    9  —    36 

1  X  10  =    10 

2X10=    20 

3  X  10  —    30 

4X10=    40 

1x11-  11 

2X  11—    22 

3X11—    33 

4X11—    44 

1X12-    12 

2  X  12  =    24 

3  X  12  =    36 

4  X  12  =    48 

5X    1-      5 

6X    1—      « 

7X1-      7 

8X    1—      8 

5  X  '  2  =    10 

6X    2=    12 

7X    2^    14 

8X    2r^    16 

5X    3=    15 

6X    3=    18 

7X    3=    21 

8X    3=    24 

5X    4=    20 

6X    4=    24 

7X    4^   28 

8X    4=    32 

5  X    6  =    25 

6X    5=    30 

IX    5=    35 

8X    5^    40 

5X    6=    30 

6  X    6  =    36 

?X    6r=    42 

8X    6=    48 

5X    7=    35 

6  X    7  =    42 

7X    1=    49 

•  8  X    7  =    56 

5X    8=    40 

6  X    8  =    48 

?X    8=    56 

8X    8=    64 

5  X    9  =    45 

6  X    9  =    54 

7X    9^r    63 

8X      9rrr      72 

5X10=   50 

6X10^=    60 

7  X  10  =    70 

8  X  10  =    80 

5  X  11  —   55 

6X11=    66 

7X11—    77 

8XH=    88 

5  X  12  =    60 

6X12=    72 

7X12:=    84 

8  X  12  =    96 

9X    1-      9 

10  x  i—  10 

11  X    1-    11 

12  X    1—    12 

9X    2=    18 

10  X    ^—   20 

11  X    2=   22 

12  X    2  =    24 

9X    3  =   27 

10  X    3  =    30 

11  X    3=    33 

12  X    3=    36 

9  X    4  ^    36 

10  X    4=    40 

11  X    4—    44 

12  X    4=    48 

9X    5=   45 

10  X    6  =    50 

11  X    5=    55 

12  X    6=    60 

9X    6=    54 

10  X    6=    60 

11  X    6  =    66 

12  X    6=r    72 

9  X    1  =    S3 

10  X    7=    70 

11  X    7=    77 

12  X    7=    84 

9X    8  =    72 

10  X    8=    80 

11  X    S=    88 

12  X    8  =    96 

9X    9=    81 

10  X    9=    90 

11  X    9=    99 

12  X    9  =  108 

9X10=    90 

10  X  10  =  100 

11  X  10  =  110 

12  X  10  =  120 

9X11—    99 

10  X  11  =  110 

11  X  11  —  121 

12X11  =132 

9  X  12  —  108 

10  X  12  ==  120 

11  X  12  —  132 

12  X  12  =  144 

NELSON'S 

COMMON-SCHOOL   ARITHMETIC. 


I.  INTRODUCTION. 

1.  Arithmetic  is  the  art  or  science  of  computing  num- 
bers. 

2.  The   theory  of  Arithmetic  treats  of  the   properties 
and  relations  of  numbers. 

3.  The  practice  of  Arithmetic  shows  the  application  of 
numbers  to  business,  the  mechanic  arts,  etc. 

4.  AEITHMETICAL  SIGNS. 

-f-,  called  plus,  is  the  sign  of  addition. 

=  is  the  sign  of  equality.  5-}-4=9,  is  read,  five  plus 
four  equals  nine. 

— ,  called  Minus,  is  the  sign  of  subtraction.  5 — 3=2, 
is  read,  five  minus  three  equals  two. 

-r-  is  the  sign  of  division.  9-r-3=3,  is  read,  nine  divided 
Ly  three  equah  three. 

.  is  the  decimal  sign.  Placed  at  the  left  of  a  number, 
it  represents  tenths,  hundredths,  etc. 

:  : :  :  is  the  sign  of  proportion.  3 : 4 : :  6 : 8,  is  read, 
three  is  to  four  as  six  is  to  eight. 

4.  4',  4",  4'",  is  read,  four,  four  prime,  four  second,  four 
third.  f 

3  (25) 


26  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

I/  is  called  the  radical  sign,  or  sign  of  square  root. 
1/4=2,  is  read  the  square  root  of  four  equals  two. 

-ty ' .  A  figure  inserted,  as  the  3,  indicates  the  root  to  be 
taken;  3,  the  cube  root;  4,  the  fourth  root. 

32.  A  small  figure  written  as  the  2  in  the  margin,  indi- 
cates that  the  number  is  to  be  raised  to  a  corresponding 
power;  2,  the  second  power;  3,  the  third  power.  This 
figure  is  called  the  index  of  the  power. 


II.  NOTATION  AND  NUMERATION. 

5.  Notation  is  the  art  of  representing  numbers  by  sym- 
bols,   called  figures    or    digits.     There    are    ten    of    these 
figures : 

0123456789 

naught,  one,     two,    three,     four,     five,      six,     seven,  eight,    nine. 

The  first  is  also  called  zero,  or  cipher. 

6.  When   a   larger   number   than   nine   is  to  be   repre- 
sented, two  or  more  figures  are  used. 

7.  Numeration  is  the  method  of  reading   these   figures 
when  arranged  to  represent   numbers.     For   this   purpose 
they  are  usually  divided  into  periods  of  three  from  the 
right. 

8.  The  first  period  on  the  right  contains  units,  tens  and 
hundreds,  thus : 

125 

bund.,  tens,  units, 

which  is  read,  one  hundred  and  twenty-five. 

The  second   period  contains   units,  tens  and   hundreds 
of  thousands,  thus : 

125  ,  000 

thous.,  hund., 

which  is  read,  one  hundred  and  twenty -five  thousand. 


\ 


NOTATION  AND  NUMERATION.  27 

The  third  period  contains  units,  tens,  and  hundreds  of 
millions,  thus : 

125,000,000. 

mills.,  thous.,  hund., 

which  is  read,  one  hundred  and  twenty-five  millions. 

The  fourth   period  contains   billions,  the    fifth   trilli)nsy 
the  sixth  quadrillions,  the  seventh  quintillions. 

RECAPITULATION. — The  first  period  is  hundreds,  the  second  thou- 
Bands,  the  third  millions,  the  fourth  billions,  etc. 

To  read  12376421007,  we  point  off  thus: 

12,376,421,097. 

Here    are    four    periods — the    fourth    is    billions.      The 
number  is  12  billions,  376  millions,  421  thousand  and  97. 

*WRITE  THE   FOLLOWING  IN  WORDS: 


1 

2 

3 

4 

15 

127027 

1464780 

27100101 

25 

184194 

1700700 

198140197 

125 

710107 

4001001 

100001009 

1125 

411001 

7119031 

600100100 

23125 

100020 

1000010 

190999009 

5 

3214100967831 

4191006848219 

WRITE  THE  FOLLOWING  IN  FIGURES: 

1.  Ten. 

2.  Seventeen. 

3.  One  hundred  and  twenty. 

4.  Three  hundred  and  twenty-four. 

5.  One  thousand  and  eighty. 

6.  One  hundred  and  twenty  thousand  five  hundred. 

7.  Three  hundred  and  ninety-seven  thousand  four  hun- 

dred and  forty-four. 

*  Exercises  under  articles  10,  11,  and  12  may  precede  these. 


28  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

8.  Twelve  millions  one  hundred  and  twenty-five  thou- 

sand one  hundred  and  one. 

9.  One  billion  one  hundred  and  one  thousand  and  one. 

10.  Thirty  millions   eighty-five  thousand    one    hundred 

and  seven. 

11.  Seventy-six  trillions  five  hundred  and  forty  billions 

one  hundred  and  ten  millions  and  sixty -seven. 

12.  Two  hundred  millions. 

13.  One  hundred  trillions. 

14.  Seven  hundred  billions  three    thousand  and   seven. 

15.  One  thousand. 

16.  One  million. 

17.  One  trillion  one  hundred  thousand. 

18.  One  billion. 

19.  Thirteen  billions  three  millions  seven  thousand. 

20.  One  hundred  and  ten  millions  and  eighty-seven. 

21.  Seventy-five  millions  six  thousand  and  nine. 

9.  FEDERAL  MONEY. 

The  name  usually  given  to  the  money  of  the  United 
States  is  Federal  Money.  It  is  reckoned  by  tens  and  hun- 
dreds. Though  there  are  various  kinds  of  gold,  silver 
and  nickel  coins,  money  is  always  reckoned  in  dollars  and 
cents,  or  dollars,  cents  and  mills. 

$  is  the  dollar  sign. 

c,  the  sign  for  cents. 

m,  the  sign  for  mills. 

$3456.87,5,  is  read,  three  thousand  four  hundred  and  fifty- 
six  dollars,  eighty-seven  cents,  five  mills. 

Mills  are  written  one  place  to  the  right  of  cents.  In 
this  book  mills  will  sometimes  be  separated  from  cents  by 
a  comma,  as  above. 

A  period  is  used  to  separate  cents  from  dollars,  the 
two  first  figures  on  the  right  being  cents  and  tens  of  cents. 


NOTATION  AND  NUMERATION.  29 

WHITE  THE  FOLLOWING  IN  WORDS ! 
12  3 

$457.  $31412.875  $507.974 

$25.25  $167.476  $1134.643 

$364.61  $365.323  $216.132 

$112.57  $4767.126  $345.815 

$3146.87  $116.254  $341.664 

$213.18  $67.141  $416.304 

$456.45  .75                                    .17c 

$1719.97  .353                                   .413m 

$7.90  $1.273                                  .674m 

$304.02  $35.144  $100.603 

$117.44  $414.124  $210.301 

$32.32  .67  c                                   .164m 

When  there  are  no  tens  of  cents,  a  cipher  is  written  in 
the  ten's  place,  as  in  304.02,  Ex.  1,  which  is  read  three 
hundred  mid  four  dollars  and  two  cents. 

Separate  the  dollars,  cents  and  mills  in  the  following, 
observing  to  write  dollars  under  dollars  and  cents  under 
cents : 


4 

5 

6 

7 

307  c 

135006  m 

3171  c 

24136  c 

M  064m 

2703  m 

1463  m 

7314  m 

32002  m 

753m 

12195m 

21364  c 

100011m 

35001  m 

3697  c 

71836  m 

1826063m 

61422  c 

2468  m 

2173  c 

116003m 

71922c 

14193  c 

1897m 

31  463  c 

143684  m 

21314  c 

1367841  m 

6897  m 

900013  c 

2136821  m 

146001  c 

18987  m 

41362  c 

21367m 

14367  c 

31464  c  13687m  98076  c  314683m 

8.  3141  c,  1386  c,  19301  c,  71432  c,  7321  c,  906301  c, 
2136  c,  4563  c,  10001  c,  30351  c,  9878  c,  45121  c,  31645  c, 
13621  c,  21001  c,  60006  c,  213621  c,  113600  c. 


30  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

9.  21360  m,  31461  m, 14565  in,  24798  m,  46503  m,  106  m, 
796543m,   45631m,  245m,   14013m,  41634m,   14563m, 
21364  m,  7856  m,  24130  m,  45103  m,  1467  m,  4136  m. 

10.  20135m,  314102m,   14163m,   213102m,    31453m, 
12063  m,  14673  m,  24135  m,  43103  m,  156307  m,  14007  m, 
34617m,  21368m,  146m,  178936m,  213641m,  14567m, 
36845m,  213146m,  145163m. 

11.  12314m,    2136m,    100135m,    21364m,    146345m, 
213984  m,  136453  m,  1467  m,  13645  m,  14107  in,  30674  m, 
1451  in,  147431  m,   14345  m,  41367  m,   1456  m,   21314  m, 
16789m,  13674m,  14683m,   1413m,    31463m,    16703m, 
4103  m,  163  m. 

12.  1456  m,  31463  m,  21456  m,  14637  m,  2136  m,  415  in, 
1467m,    23460m,    1454m,    3417m,    14567m,    14674  in, 
13654m,  14345m,  41631m,  14367m,  1456m,  145634  in, 
1456  m,  14567  in,  4453  m,  14639  m. 

13.23145m,  12346  in,  16734m,  2146m,  1678m, 
1463m,  14596m,  3045m,  163201m,  146324m,  14567m, 
14631  m,  21393  m,  7894  m,  14637  m,  21364  in,  14567  m, 
176301  m,  21367  m,  140601  m,  14563  m. 

14.  Write  in  columns,  as  before,  the  following:  Seventy- 
five  dollars,  eighty-seven  cents;  thirty-three  dollars,  sixty- 
one  cents,  five  mills;  seven  hundred  and  ninety-six  dollars 
and  sixty  cents;  five  thousand  dollars;  five  thousand  three 
hundred  and   eighteen   dollars  and  sixty-three  cents;  two 
hundred  and  fifty -six  dollars,  fourteen   cents,  four   mills; 
one  hundred  dollars,  sixty  cents,  three  mills. 

15.  Thirty-six    hundred   dollars,    seven    cents,   and   five 
mills;  eight  hundred  thousand  dollars,  forty  cents,  seven 
mills;  sixty-seven  dollars,  eighty  cents;  nine  hundred  dol- 
lars, seventy-five  cents,  six  mills;  three  hundred  and  sev- 
enty-six dollars,  six  cents,  four  mills;   sixty-four   dollars, 
eighteen  cents,  seven    mills;    fifty-nine   dollars,  six    cents, 
three  mills;  eight  hundred  dollars,  one  cent,  three  mills. 


NOTATION  AND  NUMERATION.  31 

17.  One  thousand  five  hundred  dollars,  sixty  cents, 
eight  mills;  three  hundred  and  fifty  dollars  and  six  mills; 
seventy-eight  dollars,  eighty  cents,  six  mills;  four  hundred 
fifty-seven  dollars,  sixty-four  cents,  seven  mills. 

10.  ODD  NUMBERS .* 

The  numbers  1,  3,  5,  7,  and  9  are  called  odd  numbers, 
and  every  number  which  has  one  of  these  figures  in  the 
unit's  place,  as  11,  13,  15,  is  also  an  odd  number. 

1.  Write   in   columns   all   the   odd    numbers   from  1    to 
151,  observing  to  keep  units  under  units,  tens  under  tens, 
and  hundreds  under  hundreds. 

2.  Write    in   the  same  way  all   the   odd  numbers  from 
151  to  351. 

3.  Write   in   the   same  way  all   the  odd   numbers  from 
351  to  601. 

4.  Write   in   the   same  way  all   the  odd  numbers  from 
601  to  901. 

11.  EVEN   NUMBERS. 

The  numbers  2,  4,  6,  8  are  called  even  numbers,  and 
every  number  which  has  0  or  one  of  these  as  a  unit  figure, 
is  also  an  even  number. 

5.  Write  in  columns,  as  before,  all   the   even   numbers 
from  2  to  200,  inclusive. 

6.  Write  in  the  same  way  all  even  numbers  from  200 
up  to  500,  inclusive. 

7.  Write  in  the  same  way  all  the  even   numbers  from 
500  to  800,  inclusive. 

8.  Write  in   the  same  way  all   the  even   numbers  from 
800  to  1100,  inclusive. 

The  Teacher  should  follow  these  exercises  by  others,  from  dictation, 
until  the  scholars  are  taught  to  write  any  sum  without  hesitation. 

*With  very  young  learners,  these  exercises  should  precede  those 
on  pages  26  and  27. 


32  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

12.  ADDINO. 

In  the  preceding  exercises  the  learner  unconsciously 
added  2  every  time  he  passed  from  one  number  to  another. 
In  the  following  exercises  he  will  be  required  to  add  3,  4,  5, 
etc.,  and  unite  the  results  in  the  same  way.  He  should 
observe  to  write  the  figures  in  straight  lines. 

9.  Commencing  at  1,  add  3  every  time  until  you  reach 
97,  thus:  1,  4,  7,  10,  13,  putting  the  numbers  under  each 
<5ther. 

10.  Commencing  at  1,  add  4  every  time  till  you  reach 
121. 

11.  Commencing  at  1,  add  5  every  time  till  you  reach 
161. 

12.  Commencing  at  2,  add  6  every  time  till  you  reach 
200. 

13.  Commencing  at  1,  add  7  every  time  till  you  reach 
232. 

14.  Commencing  at  1,  add  8  every  time  till  you  reach 
265. 

15.  Commencing  at  1,  add  9  every  time  till  you  reach 
307. 

16.  Add  3  to  all  the  odd  numbers  up  to  101.     For  this 
purpose  write   the  odd   numbers    on   the   left  'of  the  new 
numbers,  thus: 

1     4 
3     6 

17.  Add,  in  the  same  way,  4,  5  and  6  up  to  51. 

18.  Add,  in  the  sam'e  way,  7,  8  arid  9  up  to  51. 

19.  Add  3,  4,  5  and  6  to  all  the  «ven  numbers  up  to  50. 

20.  Acid  7,  8  and  9  to  all  even  numbers  up  to  80. 

21.  Add,  in  the  same  way,  2  to  the  following  numbers: 
19,  29,  39,  49,  59,  69/79,  89,  99,  and  add   3,  4,  5,  65 

8  and  9  in  the  same  way. 


ADDITION.  33 

22.  Add  2  to  the  following  numbers,  after  writing  them 
on  the  left:  8,  18,  28,  38,  48,  58,  68,  78,  88,  98. 

Add  3,  4,  5,  6,  7,  8  and  9  in  the  same  way. 

23.  Add  3,  4,  5,  6,  7,  8    and   9  to  the  following  num- 
bers: 7,  17,  27,  37,  47,  57,  67,  77,  87  and  97. 

24.  Add  4,  5,  6,  7,  8   and   9   to  6,  16,  26,  36,  46,  56, 
66,  76,  86  and  96. 

25.  Add  5,  6,  7,  8  and  9  to  5,  15,  25,  35,  45,  55,  65,  75, 
85  and  95. 

26.  Add  6,  7,  8  and  9  to  4,  14,  24,  34,  44,  54,  64,  74, 
84  and- 94. 

27.  Add   7,  8   and  9   to   3,  13,  23,  33,  43,  53,  63,  73, 
83  and  93. 

The  Teacher  ought  to  examine  his  scholars  on  the  terms,  signs 
and  principles  of  each  rule.  In  this  subject,  on  the  difference  be- 
tween notation  and  numeration,  how  many  figures  necessary  to 
write  one  hundred,  how  many  a  thousand,  etc. 


III.  ADDITION.* 

13.  The  method  of  uniting  two  or  more  numbers  into 
one  is  called  Addition. 


*The  Teacher  will  find  exercises  for  evening  study  at  the  end  of 
this  chapter. 

Beginners  should  not  be  allowed  to  count  on  their  fingers  or 
talk  over  the  process.  If  drilled  in  the  use  of  the  catch-figure  by 
blackboard  exercises,  they  will  not  afterward  resort  to  any  of  the 
slower  methods  of  computation.  The  use  of  the  catch-figure  is  in 
part  taught  in  exercises,  page  32  and  83.  For  the  purpose  of  drill, 
it  ought  to  be  taught  as  follows :  "7  and  9,  the  unit  figure  is  what?'' 
"6."  "17  and  9?"  "26."  "27  and  9?"  "36."  "Observe  that 
when  7  and  9  are  added,  the  unit  figure  is  6."  "47  and  9?" 
"57  and  9?"  "67  and  9?" 

Classes  should  be  exercised  in  this  way  through  all  the  combi- 
nations found  iii  the  exercises  referred  to.  » 


34  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

14.  The  one  number  is  called  the  sum,  amount,  total,  or 
footing. 

15.  The  sign  is  +,  and  is  called  phis.     When   placed 
between   two   numbers    it    indicates    that   they   are   to  be 
added   together.     3-f-2— 5,  is  read,  three  plus  two  equals 
Jive. 

16.  In  performing   operations   in  addition,  it  is  neces- 
sary to  write  the  units,  tens,  hundreds,   etc.,  of  the  one 
number  under  the  units,  tens,  hundreds,  etc.,  of  the  other, 
which  arranges  the  figures  in  one  straight  line. 

1.  To  add  together  135,  241  and  323. 

135  EXPLANATION. — Here  units  are  placed  under  units,  teng 

241  under  tens  and  hundreds  under  hundreds.  After  arranging 
323  the  figures  thus,  we  commence  at  the  right-hand  column  and 
•  ••  add  3  to  1,  which  makes  4,  to  which  add  5  and  we  have  9. 
Adding  the  tens'  column  in  the  same  way,  we  have  9 
tens,  which  we  write  in  the  tens'  place. 

Adding  the  hundreds'  column  in  the  same  way,  we  have  6,  which 
we  write  in  the  hundreds'  place. 

Find  the  sum  of  each  of  the  following  groups :  * 


2 

3 

4 

5 

6 

7 

8 

3131 

131 

211 

1534 

3143 

3141 

2131 

223 

453 

765 

1232 

2102 

5432 

1036 

115 

100 

23 

1002 

1413 

1426 

5812 

3799 

684 

999 

0 

10 

11 

12 

13 

14 

15 

211 

3145 

4512 

2131 

14132 

14413 

1613 

101 

4132 

1035 

1027 

1734 

34441 

143 

65 

1712 

4241 

1720 

4113 

41104 

1233 

*  Answers  arranged  promiscuously:  377,  9788,2989,  19979,  3768, 
9999,  6658,  8989,  3989,  8979,  4878,  89958. 


ADDITION.  35 

To  add  325,  42  and  178.  325 

42 
178 

Answer,     545 

EXPLANATION. — 1.  Placing  the  numbers  as  directed,  we  proceed 
to  find  the  amount  of  the  first  column  on  the  right:  8,  2  and  5  are 
15;  that  is,  1  ten  and  5  units. 

2.  Writing  the  5  units  under  the  units,  we  add  the  1  ten  to  the 
tens'  column. 

#.  This  one  added  to  the  7,  4  and  2  makes  14  tens  or  1  hundred 
and  4  tens. 

4.  Writing  the  4  under  the  tens,  we  add  the  1  hundred  to  the 
hundreds'  column.     „ 

5.  This  1  added  to  the  1  aM  3  in  the  hundreds'  column  makes  5 
hundreds,  which  5  we  write  in  the  hundreds'  place  and  our  work  is 
done. 

4501 
17.  To   add   4501+3213+1007+302,  we  write    3213 

them  thus:  1007 

302 


An*.  9023 
Add  together  the  following  numbers: 

18.  3478,  3167,  4199,  7854,  3456.  An*.  22154 

19.  1417,  210,  61907,  216,  3184.  An*.  66934 

20.  7894,  2176,  7,  109,  7998.  An*.  18184 

21.  376  +  100+71+416+709+317.  An*..   1989 

22.  1006+3009+79999+7098+17.  Am.  91129 

23.  316+10069+9777+307+198.  An*.  20667 

24.  789632+4+67+879002+876+970   is  how  much? 

Ans.  1670551 

25.  98632+76398+832+97+10029+97384     is     how 
much?  An*.  283372 

26.  1324+4354653+12+876+97843+68473    is    how 
much?  Am.  4523181 


36 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


27.   31465+2316532+107+3790+465321+3654563+ 

107653+23650+1007+30672+503102+21063     is    how 
much  ? 

28.18230+476+41034+9875+65432+5678+12090+ 
9387+8276+565+13654+443^how  much? 

Ans.  Sum  of  27  and  28,  7344065. 

29.  Add    together   45679+9837+18708+7967+485+ 
78963+84989+12345+7069+8090+7483+96748. 


TAKING  TWO   AND  THREE   FIGURES    AT  A  TIME. 

To  enable  scholars  to  grasp  two  and  three  figures  at  a  time,  and 
carry  them  up  as*one,  they  might  be  exercised  on  the  blackboard 
in  such  sums  as  the  following: 


136 
964 


4    3 

9    8 


4    6 

8    2 
1    3 


7  3 
1  6 
4  1 


9    2 

3  8 

4  1 


1  3 
4  5 
3  6 


693 
673 
212 


Such  exercises  ought  to  be  of  frequent  occurrence  and  scholars 
encouraged  to  answer  in  concert. 

The  answers  should  be  given  instantaneously,  naming  only  the 
unit  figure,  as  shown  in  the  column  below: 

After  writing  on  the  right  of  the  first  column  the 
figures  produced  by  pairing,  the  teacher  may  lead  the 
class  in  adding,  thus:  17  and  3?  30  and  1?  41  and 
6?  47  and  7?  54  and  1?  65  and  6?  81  and  5?  96 
and  2?  108  and  11? 

It  will  be  observed  that  the  tens  produced  iu  forming 
the  pairs  were  not  named.  The  same  course  should  be 
pursued  in  the  class,  as  the  learner  is  unconscious  of 
making  as  great  an  effort  as  he  really  does. 

When  the  ten  is  omitted  by  mistake,  attention  should 
n      be  called  to  it  by  giving  the  full  number,  as  15  or  11 

instead  of  5  or  1. 

1  The  other  columns  should  be  added  without  the  aid 

of  the  marginal  figures. 

After  thorough  drill  in  this,  the  class  should  be 
taught  to  take  three  figures  and  even  four  as  rapidly 
as  oae. 


3456  \  1 
1345  /  L 
3689  ^  9 
1563  f* 
9456  \  - 
3689  j  D 
8998  |  A 
1898  f° 
9873  )  , 
1678  f 
1684  >7 
7893  f  ' 
1453  1 
1763  / 
2195 
9876 
7897  \ 
2536  /  ° 
S529  \  - 
1438  /  f 


ADDITION.  37 

30.  Find  the  sum  of  8934,  16749,  809,  67549,  98697, 
746839,  1498,  829555,  9218967,  8347912,  968000,  74685. 
Total  of  the  preceding  two,  20758557. 

Foot  up  the  following  columns : 
31          32          33         34         35 

31645  3454  4213  1565  3654 

98760  2136  631*4  3657  1095 

36875  1364  2316  5437  9014 

57893  4633  1369  3457  6789 

14567  9897  9306  1234  9687 

34564  7879  6039  3421  5764 

46387  2164  8109  6789  1567 

93178  4163  9876  1746  9139 

78163  4569  6789  3456  1456  ' 

64518  5496  4567  1378  2345 

17514  6428  5679  5932  5432 

45678  8297  3263  4567  6542 

21364  9287  9457  1657  1395 

7198  7928  1459  6574  3642 

3165  9872  1455  5638  1365 

4124  8729  9375  4932  2315 

1345  9314  5976  1397  9365 

3146  3162  7639  9765  3510 

4165  2136  7938  3765  1096 

3216  9364  3959  1456  3765 

36.  Add  together  the  following  numbers:  313,  2109, 
6785,  2736,  798,  987,  21363,  316,  4934,  2178,  1009,  396, 
298,  2753,  607,  3145,  213,  6709,  6093,  190,  2130,  2160, 
716,  213,  9876,  45678,  2137,  2198,  9039,  6789,  3097, 
4684,  2136,  2178,  5672,  1987,  6789. 

Answers  promiscuously  arranged:  95368,  77823, 120272, 
115098,  667465,  88937,  171411. 

The  Teacher  should  not  permit  his  scholars  to  divide  these  col- 
umns when  adding,  nor  should  he  allow  them  to  resort  to  the  aid 
of  strokes  or  practice  counting  on  their  fingers. 


38 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


37 

38 

39 

40 

41 

3286 

2467 

34564 

46321 

3614 

6713 

109 

12345 

13632 

1364 

3654 

3178 

65435 

14567 

5436 

176 

145 

87654 

53678 

7835 

3976 

6178 

34564 

86367 

4678 

6345 

4156 

13682 

85432 

8793 

9823 

7532 

75671 

36457 

701 

6023 

9890 

86317 

21836 

9804 

1367 

6821 

24328 

17354 

1306 

8965 

9854 

98713 

63542 

717 

8632 

3821 

21345 

78163 

2103 

1034 

5843 

1286 

82645 

6397 

6312 

1936 

78654 

34685 

1096 

4593 

7136 

19876 

31768 

2130 

3687 

9876 

93643 

65314 

3107 

5006 

2863 

6356 

68231 

167 

7164 

123 

78397 

64037 

2109 

1763 

7436 

21602 

34685 

3678 

2139 

1567 

71346 

35962 

2176 

8236 

2563 

28653 

21363 

5432 

7860 

8432 

17648 

78636 

2137 

3613 

1345 

82351 

19854 

28639 

109 

8736 

21368 

80145 

1765 

1756 

8654 

78631 

87654 

371 

6386 

1263 

17639 

12345 

71031 

9890 

1345 

82360 

78654 

1463 

8243 

3093 

45671 

12345 

3168 

42.  Find  the  sum  of  all  the  odd  numbers  under  100. 

43.  Find  the  sum  of  all  the  even  numbers  under  100. 

44.  Find  the  sum  of  all  the  numbers  included  in  Ex. 
5,  page  31. 

45.  Find  the  sum  of  all  in  Ex.  6. 

Answers  promiscuously  arranged:  181217,  1300099, 
126362,  1325672,  136751,  143267,  52850,  2500,  2450, 
10100,  2510. 


ADDITION.  39 

46  to  49.  Find  the  sum  of  all  in  Ex.  9,  10,  11  and  12.* 
50  to  53.  Find  the  sum  of  all  in  Ex.  13,  14,  15  and  16. 
54  to  57.  Find  the  sum  of  all  in  Ex.  17,  18,  19  and  20. 
58  to  60.  Find  the  sum  of  all  in  Ex.  21,  22  and  23. 
61  to  64.  Find  the  sum  of  all  in  Ex.  24,  25,  26  and  27. 

65.  Add  together  all  the  numbers  from  300  to  320,  in- 
clusive; from  3120  to  3150,  inclusive;  from  160  to   200, 
inclusive;  from  1950  to  2000,  inclusive. 

Answers  in  direct  order:  9615,  19228,  25576,  25512, 
19200,  211800. 

17.  To  add  Federal  Money,  we  place  dollars  under  dol- 
lars, cents  under  cents  and  mills  under  mills,  and  proceed 
as  before. 

66.  What    is    the    amount    of  the    following    sums    of 
money?     $32.74,  $16.73,  $13.09,  $37.40,  $16.74,  $7.07. 

Am.  $123.77. 

OPERATION.     EXPLANATION. — The  sum  of  the  first  column  being  27 
$32.74    cents,  we  write  the  7  and  add  the  2  tens  of  cents  to  the 

16.73  tens'  column,  making  27  tens,  or  2  hundreds  and  7  tens. 
13.09    Writing  the  7,  we  add  the  2  hundreds  to  the  next,  which 
37.40    is  the  dollar  column,  and  proceed  as  in  the  above. 

16.74  The  second  column  of  cents  might  be  called  the  dime*' 
7.07    column. 

$123.77  Amount. 

67.  Find    the    amount    of    the    following:     $1708.25, 
$2076.00,  $709.07,  $109.88,    $999.87,   $370.04,   $695.83, 
$797.00,    $87.00,    $400.40,    $198.08,    $109.65,    $364.08, 
$217.00,  $364.09,  $785.66,  $699.08,  $776.08. 

*The  Teacher  can  work  most  of  these  by  Arithmetical  Progres- 
sion. As  indicated  by  the  numbers,  these  exercises  may  be  broken 
into  three  or  four  parts,  if  considered  too  difficult. 


40  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

G8.  $1670.03+81006.01+     $364.01+85432.99 

$2310.00+$1068.24+$26107. 18+82136.18 

$109.79+  $999.99+     $666.56+  $449.99 

$777.00+$7999.00+  $6666.00+86730.15 

69  70  71  72 

$987.67      $716.27     $187.20      $4519.27 
873.35       855.60      257.65      7864.20 

473.92  219.76  330.17  9510.33 

187.87  912.67  700.00  3578.84 
119.16  107.30  175.85  4875.60 
160.97  87.60  150.50  6115.90 
176.01  101.19  37.50  9885.10 
634.16  808.08  57.63  1105.75 

585.26  981.61  109.87  5760.87 
458.39  225.00  987.05  7901.57 

385.93  811.29  1285.58  7119.85 
589.38  300.92  2327.88  5006.29 
107.20  10.15  8900.17  9110.11 

70.99  106.30  209.18  4362.17 

18.18  547.67  101.01  3210.18 

1764.18  336.44  125.00  2133.64 

397.27  176.33  117.45  1364.57 
444.99  1275.84  361.45  2136.06 
222.66  666.57  217.33  1456.27 

799.88  1176.22  163.77  376.22 

73.  Find  the  amount  of  the  following  sums  of  money: 
One  hundred  and  twenty-five  dollars  and  twenty-five 
cents;  Sixty-eight  dollars  and  forty-seven  cents;  Three 
hundred  and  ten  dollars  and  eighty-seven  cents  j  Six  hun- 
dred dollars  and  seven  cents;  Four  thousand  eight  hun- 
dred and  fifty  dollars  and  eighteen  cents. 

Answers  arranged  promiscuously,  including  those  to  Ex. 
67  and  74:  $11467.06,  $64493.12,  $45178062.31,  $5954.84, 
$2624.91,  $10422.81,  $16802.24,  $97392.79,  $9457.42, 
$96394.79. 


ADDITION.  41 

74.  Add  together  the  following  amounts: 

Eighteen  thousand  one  hundred  and  forty-six  dollars; 
Seven  thousand  one  hundred  and  sixteen  dollars  and 
twenty-five  cents;  Sixty-four  thousand  one  hundred  dol- 
lars and  four  cents;  Forty-five  millions  and  one  thousand 
dollars;  Eighty-seven  thousand  seven  hundred  dollars  and 
two  cents. 

75.  A  merchant  has  29  pieces  of  silk  in  1  package,  35 
in   another,  79   in   a   third.     In   the   first  there  are  1497 
yards;  in  the  second,  2173  yards;  in  the  third,  4130  yards. 
How  many  pieces,  how  many  yards? 

SOLUTION.  29  1497 

35  2173 

79  4130 


Whole  number  of  pieces  143      Whole  number  of  yds.  7800 

*76.  A  coal  dealer  sells  1254  bushels  every  day  of  the 
week,  Sunday  excepted;  how  many  does  he  sell  in  all? 

To  THE  TEACHER. — Columns  of  fifteen  or  twenty  numbers  may 
now  be  dictated  to  classes,  the  teacher  observing  to  increase  the 
speed  of  the  scholars  at  every  effort.  The  results  may  be  called 
off  as  produced,  and  written  by  the  teacher  on  the  blackboard,  or 
the  learners  may  exchange  slates  for  examination  and  correction. 

In  this,  as  in  all  competitive  exercises,  the  teacher  should  not 
wait  until  every  member  of  the  class  has  finished  the  work;  but 
the  tardy  ones  must  not  be  overlooked,  nevertheless.  Means  should 
be  adopted  to  stimulate  them  to  greater  effort.  They  must  be 
taught  that  they  can  not  be  allowed  to  fall  behind  without  the 
risk  of  being  returned  to  a  lower  class  or  grade. 

The  teacher  probably  knows  that  to  make  boys  or  girls  reckon 
rapidly  he  must  lead;  and  to  this  end,  it  would  be  to  the  advantage 
of  both  teacher  and  pupil  if  such  exercises  as  these  were  always 
impromptu. 

*In  giving  the  answers,  the  learner  should  state  whether  it  i3 
bushels,  pounds,  etc.     The  abbreviation  Ibs.  stands  for  pounds. 
4 


42  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

77.  A    merchant   bought    9   bags    of   coffee,    each   bag 
weighing   215  Ibs.,  at  an   average   cost   of  $21.20;    what 
weight  of  coffee  did  he  buy,  and  how  much   did  he  pay 
for  it? 

78.  A   farmer  has   118   sheep,  518   hogs,  210   pair   of 
chickens,  5  plows,  6  wagons,  1  dozen  hoes,  7  horses,  10 
spades  and  12  pitchforks;  how  many  animals  has  he,  and 
how  many  agricultural  implements? 

79.  How  many  pupils  in  a  school  in  which  there  are  5 
classes,  the  first  containing  19,  the  second  28,  the  third 
32,  the  fourth  35  and  the  fifth  29  pupils  ? 

80.  Bought   three   boxes   of  oranges,  in  one  of  which 
there  were  450,  in  another  469,  in  the  last  510  oranges; 
how  many  did  I  buy? 

81.  A  man  walked  twenty-five  miles  on   the   20th   day 
of  the  month,  twenty-three  on  the  22d,  twenty-nine  on  the 
23d,  thirty-three  on  the  24th  day;   how  many  miles  did 
he  walk  altogether? 

82.  How  many  days  in  the  first  nine  calendar  months 
of  the  year  ? 

83.  Sir  Isaac  Newton  lived  85  years  and  was   born  in 
1642;  in  what  year  did  he  die? 

Answers:  143,  1429,  273,  110,  274,  1727,  7524,  1935, 
19080,  1063,  45. 

The  Teacher  should  give  numerous  exercises  besides  thesd,  and 
have  his  scholars  work  them  on  the  blackboard  before  the  class. 


HOME  EXERCISES. 

1.  Add  1,  2,  3,  4,  5,  6,  7,  8  and  9  to  2,  and  take  them  from 
the  result  again,  writing  them  out  as  below : 
2  and  1  are  3;  2  from  3  leaves  1. 
2  and  2  are  4;  2  from  4  leaves  2. 
2  'and  3  are  5;  2  from  5  leaves  3. 


SUBTRACTION.  43 

2.  Add  and  subtract  (take  from)  3  in  the  same  way. 
3    Add  and  subtract  4  in  the  same  way. 

4.  Add  and  subtract  5  in  the  same  way. 

5.  Add  and  subtract  6  in  the  same  way. 

6.  Add  and  subtract  7  in  the  same  way. 

7.  Add  and  subtract  8  in  the  same  way. 

8.  Add  and  subtract  9  in  the  sam^  way. 


IV.  SUBTRACTION. 

18.  The  process  of  finding  the  difference  between  two 
numbers  is  called  Subtraction. 

19.  This  difference  is  called  the  remainder  or  excess. 

20.  The  sign  is  — ,  and  is  called  minus.     When  placed 
between  two  numbers,  it  shows  that  the  one  on  the  right 
is  to  be  taken  from  the  one  on  the  left :  7 — 5=2,  reads, 
seven  minus  Jive  equals  two. 

1.  To  find  the  difference  between  375  and  263. 
SOLUTION. — 1.  Writing   the  small  number  under   the  375 

large  one,  units  and  tens  of  the  one  under  those  of  the  263 

other,  we  proceed  to  subtract  3  from  5,  which  leaves  2;  

this  we  write  under  the  3.  112 

2.  6  from  7  leaves  1  ;  write  it  under  the  6. 

3.  2  from  3  leaves  1.     The  remainder  is  112. 

2.  From   186436  take  165213        An*.  21223 

3.  From   786900  take  654300  132600 

EXERCISES  FOR  THE  BLACKBOARD. 

•  When  subtracting  one  figure  from  another,  the  learner  should  be 
taught  to  see  the  result,  rather  than  to  reckon  it  or  talk  over  the 
process.  This  can  be  done  by  such  exercises  as  the  following : 

Taking  a  row  of  figures  as  4903781236542,  point  to  9,  requiring 
the  class  to  give  the  difference  between  it  and  4 ;  th,en  to  0,  requir- 
ing the  difference  between  10  and  9,  etc. 


44  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

4.  From  49368282  take  15012  Am.  49353270 

5.  247896—136785  8.  66145397—  134286 

6.  716035—  15012  9.  15176482—4164271 

7.  371150—  70000  10.  37898643—  276321 

Total,  1113284  Total,  114645644 

11.  1317^3^6^87^45—    13145363435 

12.  984960997610899—771900986010098 

13.  19899799994896—  7445199821886 

Total,  238677822898021. 

When  some   of  the  figures  of  the    smaller    number   are 
greater  than  those  above  them,  we  add  ten  to  both  figures. 

14.  From  342  take  267. 

To  subtract  the  67  from  the  42  above  it,  we  add  10  to  10 

3J.  V 

both  numbers,  as  indicated  by  the  small  figures;  but  in- 
stead of  adding  to  both  numbers  in  the  same  place,  we  267 
add  10  to  the  unit  2  of  the  upper  number,  and  1  (ten  )to 
the  6  tens  of  the  lower  number.  '  ^ 

PROCESS. — 1.  We  can  not  take  7  from  the  2;  add  10  which  makes 
12;  7  from  12  leaves  5,  which  write. 

2.  Adding  1  (ten)  to  the  6  we  have  7,  which  taken  from  14,  after 
adding  another  ten  (or  one  hundred)  leaves  7. 

3.  Adding  1  (hundred)  to  the  2  we  have  3,  which  taken  from  3 
leaves  nothing.     The  remainder  is  75. 

21.  PROOF. — By  adding  the  remainder   to   the  smaller 
number,  we  should  get  a  sum  equal  to  the  larger. 

In   the   above    example    the   remainder   was    75,    the 
smaller  number  267,  which  added  together  equal   the  *?& 

larger,  342. 

*15.  1603845732164000000 

98123456789798768 


1505722275374201232  Rem. 


*The  Teacher  will  require  the  class  to  read  off  both  question  and 
answer. 


SUBTRACTION  45 

16.  From  31642789600  take  1278899765. 

Rem.  30363889835. 

17.  From  16782466987  take  123469978. 

Rem,.  16658997009. 

18.  21683673217849—1637,642178 

19.  221682178368001—1000999999 

20.  100000000000000—        1 

21.  3681068213682—  11194680 

Total,  347044269962674. 

22.  From  $1670  take  $389.27. 

OPERATION.     $1670.00 

389.27 


$1280.73  Am. 
When  writing  the  following  questions,  be  particular  to 

To  THE  TEACHER. — Mental  exercises  in  adding  will  be  found  a 
good  means  of  cultivating  the  retentive  faculty  for  business  pur- 
poses. Such  exercises  should  not  consist  of  single  digits  nor  of 
fractions  with  terms  made  up  of  single  digits,  but  of  numbers  such 
as  the  following: 

27  and  64  are  how  much?  To  add  these  numbers  the  tens  should 
be  taken  first:  27  and  60^87  and  4=91. 

$3.27  and  $1.25  are  how  much?  .  327  and  120=447  and  5=452. 

COMPETITIVE    EXERCISES 

In  addition  might  come  in  here,  and  be  introduced  at  intervals 
throughout  the  course.  A  problem  being  written  on  the  black- 
board, or  dictated  to  the  class,  scholars  should  be  required  to  hand 
in  their  slates  in  the  order  in  which  they  obtain  the  results,  when, 
the  teacher  would  number  them,  "1,  2,  3,  wrong,  4,  5,"  etc.,  calling 
the  name  of  the  competitor  in  each  case  and  returning  the  slatei 
Such  exercises  should  be  conducted  by  the  teacher  with  celerity,  so 
that  at  a  single  glance  he  can  tell  whether  the  learner  has  the  cor- 
rect result.  Fifteen  minutes  will  be  found  sufficient  time  to  devote 
to  such  exercises  at  once. 


46  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

arrange    dollars    under    dollars    and    cents    under    cents. 
When  there  are  no  cents,  write  two  ciphers  in  their  place. 

23.  $10067.89— $2141.98      27.       $60000.00— $4670.87 

24.  $15070.14—  6160.47      28.       $23678.45—  4101.00 

25.  $1001000.     —         1.86      29.  $1006812.00—  3178.59 

26.  $6743147.     —  .97      30.     $678997.00—    210.99 
Total,  $7760979.75.  '  Total,  $1757326.00. 

31.  $710356.87— $14683.29  35.     $68750.37— $1416.44 

32.  $370968.     —  17987.77  36.     $71000.90—       87.50 

33.  $478979.     —  14780.99  37.  $100000.     —          .87 

34.  $100000.     —       374.66  38.     $61987.15—  .99 
Total  dif.,  $1612477.16.             Total  dif.,  $300232.62. 

^39.  What  is  the  sum  and  difference,  when  added  to- 
gether, of  $36748..94  and  $10968.75. 

40.  Borrowed  of  A,  at  different  times,  $146.87,  $6740.18, 
$310.75,  and  have  paid  him  $10.00,  $450.18  and  $61.14; 
how  much  do  I  owe? 

41.  Out  of  5  hogsheads  of  sugar  containing  5761  Ibs., 
I   sold  3,  containing  1114  Ibs.,  1311   Ibs.  and  1001  Ibs.; 
how  much  was  left? 

42.  After  selling  1347  Ibs.  of  sugar  from  3  hogsheads, 
each  containing  1000  Ibs.,  how  much  was  left? 

EXERCISES  IN  TAKING  THE  COMPLEMENT  OB  "MAKING 
CHANGE." 

To  THE  TEACHER. — Taking  $1  as  the  complete  number,  require 
the  complement  of  25  c,  27  c,  30  c,  35  c,  etc.  It 'will  be  found  that 
the  complement  of  the  teens  is  in  the  80s,  of  the  20s  in  the  70s,  of 
the  30s  in  the  60s,  of  the  40s  in  the  50s,  of  the  60s  in  the  30s,  of 
the  70s  in  "the  20s,  etc.  "54?"  Aris.  46.  35?  Ans.  65. 

Taking  $2,  $3  or  $5,  the  exercise  may  be  practiced  in  the  same 
way. 

*Give  the  denomination  of  the  answer,  whether  it  be  dollars, 
pounds,  etc. 


SUBTRACTION.  47 

43.  A  merchant,  owns  goods  to  the  amount  of  $3147, 
and  lands  to  the  amount  of  $2107,  and  is  indebted  $1400 : 
to  A  $200,  to  B  $340  and  to  C  $860;  what  is  the  amount 
of  his  net  capital  ? 

44.  A  merchant  sells  goods  for  another  to  the  amount 
of  $4374.23,  and  is  to  receive  $43.75  for  his  trouble,  be- 
sides  the   expenses   of  freight,  etc.,  which  was    $125.15 ; 
how  much  should  he  return  to  his  principal? 

45.  What   is    the    difference    between    1856   and   1798? 
When  was  the  individual   born  who  died  in  1857  at  the 
age    of  45    years?     When   will    the  work    be    completed 
which  was  commenced  in  1855  and  was  to  take  eighteen 
years  ? 

Answers   to    the   last   seven:    420533,   58,  1812,    2454, 
1873,  1653,  7349788,  2335,  667648,  410533 .* 

*Give  the  denomination  of  the  answer,   whether  it  be  dollars, 
pounds,  etc. 

HOME  EXERCISES/}- 


1.  30+25=? 

14.  78+65=? 

27.  94+27=? 

2.  27+40=? 

15.  59+63=? 

28.  79+86=? 

3.  26+30=? 

16.  72+48=? 

29.  56+65=? 

4.  43+27=  ? 

17.  37+75=? 

30.  33+63=? 

6.  29+12=? 

18.  59+76=? 

31.  98+63=? 

6.  54+16=? 

19.  85+64=? 

32.  27+79=? 

7.  33+17=? 

20.  78+77=? 

33.  56+65=? 

8.  55+75=? 

21.  99+37=? 

34.  73+64=? 

9.  47+43=? 

22.  55+66=? 

35.  55+68=? 

10.  82+12=? 

23.  37+74=? 

36.  97+63=? 

11.  95+15=? 

24.  65+37=? 

37.  64+67=? 

12.  34+45=? 

25.  59+63=? 

38.  73+67=? 

13.  29+64=? 

26.  37+65=? 

39.  45+63=? 

t  To  be  written  on  the  slate  at  homeland  the  operations  recited  in 
Bchool. 


48  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

40.  27-f-33=?  60.  59+16=?  80.  97+63=? 

41.  95+64=?  61.  27+17=?  81.  84+37=? 

42.  84+33=?  62.  86+59=?  82.  29+18=? 

43.  27+69=?  63.  75+38=?  83.  63+97=? 

44.  55+69=?  64.  64+55=?  84.  25+88=? 

45.  78+73=?  65.  29+33=?  85.  36+54=? 

46.  65+63=?  66.  36+93=?  86.  97+79=? 

47.  73+39=?  67.  74+67=?  87.  88+69=? 

48.  55+97=?  68.  36+56=?  88.  53+76=? 

49.  77+67=?  69.  75+27=?  89.  95+84=? 

60.  59+63=?  70.  33+13=?  90.  67+96=? 

61.  78_|_64=?  71.  29+19=?  91.  44+39=? 

62.  59+33=?  72.  98+29=?  92.  29+88=? 

63.  98+97=?  73.  29+35=?  93.  74+86=? 
54.  65+36=?  74.  18+62=?  94.  49+57=? 
65.21+17=?  75.79+84=?  95.64+88=? 

66.  76+93=?  76.  56+33=?  96.  98+77=? 

67.  54+64=?  77.  49+54=?  97.  55+76=? 

68.  79+98=?  78.  59+39=?  98.  74+84=? 

69.  63+64=?  79.  28+97=?  99.  85+99--^? 

Write  the  multiplication  table  as  follows: 

1.  2  times  1  or  once  2  is  2. 
2  times  2  are  4. 

2  times  3  or  3  times  2  are  6. 
2  times  4  or  4  times  2  are  8. 
Continue  this  to  12. 

2.  Write  the  3  times  table  to  12  in  the  same  way. 

3.  Write  the  4  times  table  to  12  in  the  same  way. 

4.  Write  the  5  times  table  to  12  in  the  same  way. 
6.  Write  the  6  times  table  to  12  in  the  same  way. 

6.  Write  the  7  times  table  to  12  in  the  same  way. 

7.  Write  the  8  times  table  to  12  in  the  same  way. 

8.  Write  the  9  times  table  to  12  in  the  same  way. 

9.  Write  the  10  times  table  to  12  in  the  same  way. 

10.  Write  the  11  times  table  to  12  in  the  same  way. 

11.  Write  the  12  times  table  to  12  in  the  same  way. 


MULTIPLICATION. 


IV.  MULTIPLICATION 

22.  Multiplication  is  a  short  method  of  adding,  when 
the  same  number  has  to  be  repeated  any  number  of  times. 
X  is  the   sign.     3X^—18,  reads,  three   times   six   equals 
eighteen. 

1.  To  find  the  sum  of  123+123+123,  by  additions  we 
would  enter  the  three  amounts  as  before,  and  add  for  the 
result. 

In  multiplication,  we  multiply  each  figure  of  the  num-  123 

ber  to  be  increased  by  the  number  which  indicates  3 

liow  often  the  repetition  is  to  be  made,  thus:  —  .  — 

3  times  3  are  9;  put  9  in  the  unit's  place.  369 

3  times  2  are  6;  put  6  in  the  ten's  place. 

3  times  1  are  3,  which  put  in  the  hundred's  place.  The  result  is 
869,  as  it  would  have  been  by  addition. 

TERMS. 

23.  The  number  to  be  multiplied  is  called  the  multipli- 
cand, the  number  by  which  it  is  multiplied,  the  multiplier, 
and   the    number  produced   by   multiplying,   the  product. 
The  multiplicand  and  multiplier  are  also  called  factors. 

T,  (123  Multiplicand. 

FACTORS  |     jj 


369  Product. 
2.  To  find  the  product  of  1496  by  7. 

Here  we  say  7  times  6  are  42;  write  2  under  the  7.  1496 

Then  7  times  9  are  63,  and  the  4  we  carried  make  67;  7 

write  7  and  carry  6.     7  times  4  are  28  and  G  are  34;  - 

write  4  and  carry  3.     7  times  1  are  7  and  3  are  10.  10472 

Ans.  10472 


50 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


3.  2146X2=  4292 

4.  3178X3=  9534 

5.  4167X4=16668 

9.  5189X5=? 

10.  7864X6=? 

11.  2875X7=? 

Sum,  93254. 


6.  21007X5=? 

7.  31497X<5=? 

8.  17843X7=? 

Sum,  418918. 

12.  41679X  8=? 

13.  98765  X   9=? 

14.  73149X12=? 

Sum,  2100105. 


Observe  to  point  off  the   cents  in  the  products  of  the 
following : 


15.  $21.37X7=? 

16.  $117.49=8=? 

17.  $317.00x0—? 
Amount,  $3942.51. 

21.  $678.39X11=-? 

22.  $467.28X12=? 

23.  $999.99X  9=? 
Amount,  $22069.56. 

27.  $47.531  X   9=? 

28.  $716.145X11=?' 

29.  $9871.321x12=? 
Amount,  $126761.226. 

33.  $9057.179X12=? 

34.  $7898.796X   9=? 

35.  $5970.463X11=? 
Amount,  $245450.405. 

39.  $7161.213x11—? 

40.  $1409.796X12=? 

41.  $9393.678  X   9=? 

42.  $4131.196X10=? 

Amount.  $221545.957. 


18.  $10.73X9—? 

19.  $117.07X6=? 

20.  $307.49x7=? 
Amount,  $2951.42. 

24.  $671.49X10=? 

25.  $857.37X11—? 

26.  $1096.49X12=? 

Amount,  $29303.85. 

30.  $1670.053x7=? 

31.  $2199.989X9=? 

32.  $7186.739x8=? 

Amount,  88984.184. 

36.  $793.179X9=? 

37.  $987.970X8=? 

38.  $213.219X7=? 
Amount,  $165349.04. 

43.  $3714.291  X^—? 

44.  $7965.379X8=? 

45.  $3768.219x7=? 

46.  $  419.367X6=? 
Amount,  $126045.386. 


MULTIPLICATION.  51 

47.  2785X357. 

We  have  here  three  multipliers — seven,  fifty  and  three 
hundred. 

2785X7=                  19495  19495 

2785X5  tens=          13925  tens,  or  139250 

'  2785x3  hundreds=  8355  hundreds,  or  835500 


Total  products,  994245 

This  operation  is  contracted  by  arranging  the  figures          2785 
as   in   the  margin,  and   writing  the  first  figure  of  the  357 

products  of  the  units  in  the  unit's  place  and  the  others 
to  the  left,  of  it;  the  first  figure  of  the  product  of  the 


tens  in  the  ten's  place,  or  under  its  own  multiplier,  5;      oore 
and  the  first  figure  of  the  product  of  the  hundreds  in  the 
hundred's  place.  994245 

Either  factor  may  be  used  as  a  multiplier  in  the  fol- 
lowing exercises? 

48.  3170X  178=?  51.  2896x6789=? 

49.  6184X1794=?  52.  7109x9998=? 

50.  3867X3784=?  53.  2345x3979=? 
Total,  26291084.  Total,  100067481. 

To  THE  TEACHER.  —  Blackboard  exercises  in  concert  may  be  given 
in  the  following  manner: 

Writing  a  line  of  figures,  379463875426,  the  teacher  would  lead 
by  multiplying  the  second  by  the  first,  the  third  by  the  second, 
the  fourth  by  the  third,  etc.,  without  speaking  the  process.  Pointing 
to  7,  he  would  say  21;  to  9,  63;  to  4,  36,  etc. 

To  instruct  in  "carrying,"  the  same  line  may  be  used  by  point- 
ing to  the  third  figure  and  performing  the  following  operation 
mentally:  3X"-f  9=30.  Pointing  to  4,  he  would  say  67;  to  6,  42; 
to  3,  27. 

To  produce  rapidity  of  thought  and  action,  exercises  of  this  kind 
ought  to  be  frequent,  and  the  teacher  should  lead,  taking  care  that 
the  whole  class  follows. 

Such  exercises  as  these  may  be  profitably  continued  throughout 
the  entire  course  of  study. 


52 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


54.  6789X2164=? 

55.  1578X  753=? 

56.  9409X6781=? 

57.  2783X4679=? 

Total,  92703716. 

62.  420001000 
109608 


58.  8976X7659=? 

59.  3968X6483—? 

60.  7689X2197=? 

61.  6784X7898=?. 

Total,  1649444493. 

109608 
420001000 


3360008000 
2520006 
3780009 
420001 


109608 
219216 
438432 

Product,  46035469608000 


Product,  46035469608000 

The  multiplier  of  the  ten's  place  in  the  first  operation 
being  0,  we  passed  it,  and  multiplied  by  the  6  hundreds. 
In  the  second  operation  we  passed  the  ten's,  hundred's 
and  thousand's  places  for  the  same  reason.* 


63.  12346X30010=? 

.  64.     7684X10900=? 

65.     6787  X  3009=? 

Total,  474681143. 

69.  2000X  7010=? 

70.  3160X10096=? 

71.  2178X90909=? 

Total,  243923162. 


66.  4967X  6007=? 
67.. 5896X900707=? 
68.  7649X  66080=? 

Total,  5845851161. 

72.  1009x90910=? 

73.  21678X21006=? 

74.  31784X  7009=? 

Total,  769870314. 


24.  To  multiply  ~by  10,  100,  1000,  etc.,  we  have  simply 
to  annex  as  many  ciphers  to  the  multiplicand  as  there 
are  in  the  multiplier. 


*If  the  learner  will  simply  observe  to  write  the  first  figure  of  each 
product  under  its  own  multiplier,  he  will  have  no  difficulty  in  mul- 
tiplying where  there  are  ciphers.  For  instance,  the  first  figure  of 
the  product  by  2,  in  the  second  example,  is  immediately  under  the  2. 


MULTIPLICATION.  53 

35X10=350. 
Proof  35 

10        Ten  times  5  are  50,  and  10  times  3  are  30  and  5 

•     are  35,  making  350. 

350 

75.  1G5X     10=     1650  78.     413X     10=? 

76.  165X  100=  16500  79.  1716X   100=? 

77.  165x1000=165000  80.  9417x1000=? 

Total,  9592730. 

81.  374X     100=?  84.  9361 X     10=? 

82.  268X  1000=?  85.  7342X   100=? 

83.  189X10000=?  86.  8654x1000=? 

Total,  2195400.  Total,  9481810. 

25.  To  multiply  dollars,  cents  and  mills,  we  remove 
the  decimal  point  to  the  right. 

87.  What  is  the  product  of  $279.373  by  10? 

Am.  $2793.73. 

EXPLANATION. — By  multiplying  the  mills  by  10  we  make  them 
cents,  by  multiplying  the  cents  by  10  we  make  them  dimes,  and  by 
multiplying  the  dimes  by  10  we  make  them  dollars. 

88.  $145.373  XlOO=$14537.3,  or  14537  dollars  and  3 
dimes  or  30  cents. 

89.  $356.14,5 X  10=?      92.     $317.98,7x  100=? 

90.  $178.91,3X100=?      93.     $679.97,6X     10=? 

91.  $463.97,8X100=?      94.  $7193.44,5x1000=? 

Amount,  $67850.55.  Amount,  $7232043.46. 

95.  $713.71,4X  100=?  99.  $131.71,2x1000=? 

96.  $165.79,3X1000=?  100.  $724.26,8 X  100=? 

97.  $786.47,5X     10=?  101.  $413.16,4X     10=? 

98.  $130.14    X  100=?  102.  $236.21    X  100=? 
Amount,  $258043.15.  Amount,  $231891.44. 

To  THE  TEACHER. — Mental  exercises  on  this  subject  should  suc- 
ceed these  written  ones. 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


HOME  EXERCISES. 

I.  Commencing  at  13,  write  the  2  times  table  to  19  in 
this  way : 

2  times  13  are  26;  13  times  2  are  26. 
2  times  14  are  28 ;  14  times  2  are  28. 

2.  Write  the  3  times  table  in  the  same  way. 

3.  Write  the  4  times  table  in  the  same  way. 

4.  Write  the  5  times  table  in  the  same  way. 

5.  Write  the  6  times  table  in  the  same  way. 

6.  Write  the  7  times  table  in  the  same  way. 

7.  Write  the  8  times  table  in  the  same  way. 

8.  Write  the  9  times  table  in  the  same  way. 

9.  Write  the  13  times  table  to  9  as  follows: 

13  times  2  are  26;  2  times  13  are  26. 

13  times  3  are  39;  3  times  13  are  39. 

10.  Write  the  14  times  table  in  the  same  way. 

II.  Write  the  15  times  table  in  the  same  way. 

12.  Write  the  16  times  table  in  the  same  way. 

13.  Write  the  17  times  table  in  the  same  way. 

14.  Write  the  18  times  table  in  the  same  way. 

15.  Write  the  19  times  table  in  the  same  way. 

1.  Write  the  2  times  table  to  19  as  follows: 

2  times  3  are  6;  2  times  13  are  26. 
2  times  4  are  8;  2  times  14  are  28. 
2  times  5  are  10;  2  times  15  are  30. 

2.  Write  the  3  times  table  in  the  same  way. 

3.  Write  the  4  times  table  in  the  same  way. 

4.  Write  the  5  times  table  in  the  same  way. 

5.  Write  the  6  times  table  in  the  same  way. 

6.  Write  the  7  times  table  in  the  same  way. 

7.  Write  the  8  times  table  in  the  same  way. 

8.  Write  the  9  times  table  in  the  same  way. 


MULTIPLICATION.  55 

EXERCISES    IN  MULTIPLYING  THE  "  TEENS."* 

100.  2357X13  and  14?  104.  7890x10  and  18? 

101.  5398X15  and  16?  105.  2164x17  and  16? 

102.  6532X17  and  18?  10(1.  3165x16  and  15? 

103.  7654X17  and  19?  107.  2137x15  and  14? 
Total  products,  735141.  Total  products,  523430. 

108.  $45.67X16  and  15?  112.  $76.54   x!6  and  17? 

109.  §14.59X15  and  14?  113.  §57.352x19  and  15? 

110.  §23.08X13  and  14?  114.  §67.185x18  and  17? 

111.  §21.87X17  and  19?  115.  §45.375x16  and  18? 
Total  products,  §3249.36.        Total  products,  §8370.013. 

116.  §137.67   Xl7andl8?  120.     §43.165X10  and  16? 

117.  §216.031X15  and  18?  121.  §933         Xl8andl5? 

118.  §131.75   Xl6andl6?  122.     §61.751x16  and  17? 

119.  §231.35   Xl7andl9?  123.  §311.155x14  and  13? 
Total  prod.,  §84595.96.  Total  prod.,  §42738.743. 

f       PRINCIPLES  OF  MULTIPLICATION. 

26.  When  two  numbers  are  to  be  multiplied  together, 
we   use  for  the  multiplier   that  which  will    produce   least 
figures   in  the  operation.     This  will   be   accomplished  by 
selecting  the  smaller  number,  except  where  there  are  many 
ciphers,  as  in  Ex.  62. 

27.  If  a  number  of  articles  and  the  price  of  one  article 
be  multiplied  together,  the  product  will  be   the  price  of 
all  at  the  same  rate. 

3  yards  of  muslin  at  20  cents. 

20X'3=60  cents. 

If  the  price  of  one  be  in  cents,  the  price  of  all  wilf  be 
in  cents.     If  in  dollars,  the  price  of  all  will  be  in  dollars. 

*  Every  boy  designed  for  business  pursuits  ought  to  commit  to 
memory  the  multiplication  table  up  to  19  times,  inclusive. 


56  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

28.  The  number  of  articles  contained  in  any  box,  bale, 
package,  etc.,  multiplied  by  the  number  of  boxes,  bales, 
etc.,  each  containing  a  like  number,  will  give  the  number 
of  articles  in  all. 

In  a  box  there  are  30  articles;  how  many  in  20  such 
boxes? 

30X20=600  articles. 

29.  A  number  multiplied  by  itself  is  squared,  or  raised 
to  the  second  power,  and  the  second  power  multiplied  by 
the   same   number  is  cubed,  or  raised  to  the  third  power. 
The   sign   is   a  small    figure  on  the  right  of  the  number, 
thus,  5*,  which   indicates  that  5   is   to  be   raised   to   the 
fourth  power,  and  is  equal  to  5x5X^X5,  or  625. 

30.  Feet  multiplied   by  feet  and    yards   multiplied   by 
yards  produce  square  feet  and  square  yards. 

12  feetXl2  feet=144  square  feet. 

31.  Any  number  of  feet  multiplied  by  the  number  of 
inches  in  one  foot  will  give  the  number  of  inches  in  all 
the  feet.     Pounds  multiplied  by  the  number  of  ounces  in 
one  pound  will   give   the   number   of  ounces  in  all   the 
pounds,  and  so  with  numbers  of  any  other  denomination. 

How  many  inches  in  37  feet? 

37X12=444  inches. 

1.  What  is  the  price  of  37  bushels  of  corn  at  37  cents 
per  bushel? 

2.  What  should  I  pay  for  357-  yards  of  broadcloth  at 
$2.?5  per  yard? 

3.  Find  the  cost  of  325  acres  of  land  at  $57  per  acre. 

4.  In  320  bales  of  cotton  there  are  460  Ibs.  each ;  how 
many  in  all? 

5.  In   557   pieces   of  muslin   there   are   35  yards  each; 
how  many  in  all? 


MULTIPLICATION.  57 

6.  A  ship  laden  with  flour  has  7950  barrels  on  board, 
and  in  each  barrel  there  are  196  Ibs.;  how  many  pounds 
in  all? 

7.  In   a  bushel   of  dried   apples   there  are  25  pounds; 
how  many  are  there  in  37  bushels? 

8.  A  barrel  of  flour  weighs  196   pounds;    what  is  the 
weight  of  325  barrels? 

9.  What  will   be   the  weight   of  134  bushels   of  wheat 
when  60  Ibs.  are  allowed  to  the  bushel? 

10.  Find  the  cost  of  379  boxes  of  cheese,  each  of  which 
weighs  22  pounds,  at  25  cents  a  pound. 

11.  A  box  of  buttons  contains  a  gross;  how  many  but- 
tons are  there  in  59  boxes? 

12.  A  merchant  sold  135  barrels  of  flour  at  $6.75  apiece, 
and  with  part  of  the  money  bought  369  bushels  of  coal 
at  25  cents;  how  much  money  had  he  left? 

13.  In  236  yards  how  many  inches? 

14.  How  many  quarts  are  there  in  27  bushels? 

15.  At  23  cents  a  quart,  how  much  money  can  be  real- 
ized on  18  bushels  of  strawberries,  allowing  one  quart  for 
loss  in  measuring? 

16.  A  huckster  bought  two  barrels  of  apples,  each  con- 
taining 3  bushels,  at  $6   a   bushel,  and  sold  them  at  87 
cents  a  half  peck ;  allowing  half  a  peck  for  loss  in  meas- 
uring, did  he  gain  or  lose?  and  how  much? 

17.  There  are  12  inches  in  a  foot  and  3  feet  in  a  yard; 
how  many  inches  are  there  in  357  yards? 

18.  How  many  quarts  of  wine  are   there   in   6  barrels, 
each  of  which  contains  42  galls.?   how  many  in  35  bar- 
rels? how  many  in  163? 

Answers  without  their  denominations:  13225,  864,  819, 
8496,  208450,  8496,  8040,  925,  98175,  21397,  18525, 
147200,  63700,  19495,  1558200,  489,  1369,  12852,  34272. 


58  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

19.  Find  the  cost  of  117  bushels  of  apples  at  35  cents 
a  peck. 

20.  Find  the  cost  of  237  bushels  of  potatoes  at  42  cents 
a  peck. 

21.  Bought  46   horses   at  §125  apiece,  and   sold   them 
for  85900;  did  I  lose  or  gain,  and  how  much? 

22.  At  5  cents  apiece,  what  will  22  gross  of  eggs  cost? 

23.  In  15  acres  how  many  square  yards? 

24.  Find  the  cost  of  35  barrels  of  molasses  at  33  cents 
a  gallon,  each  barrel  containing,  on  an  average,  41  galls. 

25.  In  257  cords  of  wood  how  many  solid  feet? 

26.  How  many  bushels  of  coal  in  17  wagons,  each  car- 
rying 50?  and  what  will  be  the  cost  at  $17  a  load? 

27.  How  many  yards  are  in  a  box  of  silk  containing  35 
pieces,  each  piece  measuring  52  yards? 

28.  In   a  case   of  muslin   there   are  45  pieces,  each  31 
yards;  what  will  be  the  cost  of  it  at  55  cents  a  yard? 

29.  A   bushel   of  hemp-seed   weighs  44  pounds;    what 
will  be  the  weight  of  137  bushels? 

30.  How  many  leaves  are  in  67  reams  of  paper? 

31.  How  many  half-pence  are  in  527  pounds  sterling? 
Answers  promiscuously  arranged:  72600,  16380,  5175, 

150,   15840,  39816,  47355,  64320,  76725,   14450,   1820, 
32896,  6028,  252960. 


HOME  EXERCISES. 


1.     3+  4—  2=1 
2.    5—  3+  7=? 
3.     9><  4_  3=? 
4.     7><  6+  8=? 
6.     6+  9X  6r=? 
6.    9  —  8—  1=1 

7.     7+  3X  6=? 
8.     9-f-  8+  3=? 
9.     9><  7_  4^? 
10.     5+  5—  3=? 
11.     8X  4+  2=1 
12.    9—  3—  2=1 

13.  16+12—  5=,? 
14.  23-}-20—  1=? 
15.  19—16+  1=1 
16.  27—12-1-  8r=? 
17.  29+  2—  7=1 
18.  40+54—  3—? 

MULTIPLICATION. 


19.  35— 15X  6=? 

20.  64 -fSO—  2=? 

21.  97—1?,+  7=? 

22.  55+43—  8=? 

23.  63+65+  2=1 

24.  19+  6—  5=? 

25.  29+  6—  5=? 

26.  35+25+15=? 

27.  18+22—10=? 

28.  54+16+10=? 

29.  93+17—20=? 

30.  44+16+32  =  ? 

31.  12+18+33=? 

32.  27+33+54=? 

33.  44+16—15=? 
84.  19—12+28=? 
36.  55+44+21=? 

36.  23+60+47=? 

37.  64+15+36=? 

38.  24+46+33=? 

39.  56+44+71=? 

40.  19+33+21=? 

41.  77+44+23=? 

42.  45+31+55=? 

43.  37+13+34=? 

44.  66+14—12=? 

45.  44+76+33=? 

46.  92+18—10=? 

47.  73+10+67=? 

48.  21+39+22=? 

49.  17+63+14=? 


50.  29+33+19=? 

51.  3X15+15=? 

52.  7+16X  3=? 

53.  57— 17X  "^? 

54.  19+11X  6=? 

55.  45+15X  9=? 

56.  84+26X12=? 

57.  21+18+15=? 

58.  45+16—11=? 

59.  73+13—16=? 

60.  55+45+35=? 

61.  79+60+30=? 

62.  24+71+  5=? 

63.  39+11—10=? 

64.  99—19+12=? 

65.  77+79—14=? 

66.  85+96—11=? 

67.  77+63—20=? 

68.  45+65—22=? 

69.  33+73—20=? 

70.  97+93+50=? 

71.  79+91+93=? 

72.  56+64+49=? 

73.  73+65+37=? 

74.  93+65+35=? 

75.  97+63+47=? 

76.  45X  3+45=? 

77.  35X  4+30=? 

78.  29X  6—14=? 

79.  99X  3—27=? 

80.  84— 14X  7=? 


81.  75X  3—  6=? 

82.  36X  4—14=? 

83.  56X  3+50=? 

84.  75— 16X  6=? 

85.  35X12—20=? 

86.  42X10+22=? 

87.  57X12—14=? 

88.  45—15X30=? 

89.  64X  5—20=? 

90.  18X  6—28=? 

91.  36X12+72=? 

92.  42X12+12=? 

93.  35X20+12=? 

94.  64X32—14=? 

95.  33X21—12=? 

96.  14X12—13=? 

97.  12+33—  6=? 

98.  75X16—  4=? 

99.  39X12+  4=? 

100.  56—16+  6=? 

101.  35X15X  6=? 

102.  35—22X12=? 

103.  64+22—10=? 

104.  75X97—37=? 

105.  64X14—  4=? 

106.  36—16X96=? 

107.  55—23X20=? 

108.  49—19X84=? 

109.  64X45—15=? 

110.  97—67—17=? 

111.  37+16—  6=? 


1.  Write  the  2  times  table  to  9  as  follows: 

2  times  2  are  4 ;  2  is  contained  in  4,  2  times. 
2  times  3  are  6;  2  is  contained  in  6,  3  times. 
2  times  4  are  8;  2  is  contained  in  8,  4  times. 

2.  Write  3  times  table  to  9  as  above. 

3.  Write  4  times  table  to  9  in  the  same  way. 

4.  Write  5  times  table  to  9  in  the  same  way. 
6.  Write  6  times  table  to  9  in  tlie  same  way. 


60  NELSON'S  COMMON-SCHOOL  APITHMETiC. 

6.  Write  7  times  table  to  9  in  the  same  way. 

7.  Write  8  times  table  to  9  in  the  same  way. 

8.  Write  9  times  table  to  9  in  the  same  way. 

1.  Write  the  division  table,  as  below,  to  2  in  19: 

2  in  2,  1  time. 

2  in  3,  1  time  and  1  left. 

2  in  4,  2  times. 

2  in  5,  2  times  and  1  left. 

2.  Find  how  often  3  is  contained  in  numbers  from  3  to  29. 

3.  Find  how  often  4  is  contained  in  numbers  from  4  to  39. 

4.  Find  how  often  5  is  contained  in  numbers  from  5  to  49. 

5.  Find  how  often  6  is  contained  in  numbers  from  6  to  59. 

6.  Find  how  often  7  is  contained  in  numbers  from  7  to  69. 

7.  Find  how  often  8  is  contained  in  numbers  from  8  to  79. 

8.  Find  how  often  9  is  contained  in  numbers  from  9  to  89. 


V.  DIVISION. 

32.  Division  is  the  method  of  calculation  used  to  sep- 
arate numbers  into  equal  parts. 

33.  Division  may  be   short  or   long.     It  is  short  when 
the  process  of  finding  the  product  and  remainder  is  per- 
formed mentally,  and  long  when  the  process  is  written. 

34.  The  sign  is  -f-,  which  placed  between  two  numbers 
indicates  that  the  one  on  the  left  is  to  be  divided  by  the 
one  on  the  right. 

6-^3,  reads,  six  divided  by  three. 

Division  is  also  indicated  by  a  curved  line  between  the 
numbers,  thus:  3)6;  and  by  a  straight  line,  with  one 
number  above  and  the  other  below,  as  |,  called  the  frac- 
tional form.  The  period  is  also  used  to  indicate  division. 
.5  shows  that  5  is  divided  by  10. 

35.  TERMS. 
The  number  divided  or  to  be  divided  is  the  dividend. 


DIVISION.  61 

The  number  by  which  the  division  is  performed  or  to 
be  performed  is  the  divisor. 

The  number  which  shows  how  many  times  the  divisor 
is  contained  in  the  dividend  is  the  quotient. 

The  number  left  after  dividing,  the  remainder. 

Divisor,  3)16784  Dividend. 


Quotient,  5594  —  2  Remainder. 

SHORT  DIVISION. 

1.  To  divide  738  by  3. 

3)738  EXPLANATION.  —  1.  Commencing  at  the  left,  we  find 

-  how  often  3  is  contained  in  7  hundred,  which  is  2  hun- 
246  dred  times,  with  a  remainder  of  1  hundred.  The  2 

we  write  in  the  hundreds'  place. 

2.  This   1   hundred,  with   the  38,  gives  a   remainder  of  138   to 
be  divided.     To  divide  this,  we  consider  the   13   as  tens.     In   13 
tens  3  is  contained  4  times,  and  1  ten  left  as  a  remainder  ;  so  we 
write  the  4  in  the  tens'  place. 

3.  This  1   ten,   with    the  remaining  8,  gives   18   to   be    divided. 
In  18,  3  is  contained  6  times,  which,  being  written  under  the 

8,  gives  the  result,  246. 

Until  he  becomes  familiar  with  the  process,  the  learner 
may  write  the  remainders  in  small  figures,  as  in  the  fol- 
lowing example. 

2.  8)1    3  50  27  36  48    3  37  56    1  *7 


163460470    2—1 

EXPLANATION.  —  1.  Commencing  at  the  left  we  find  how  many 
times  8  is  contained  in  13.  The  answer  being  1  time,  with  a  re- 
mainder of  5,  we  write  the  1  under  the  13  and  the  5  before  the  0. 

2.  This  5  taken  with  the  0  makes  50,  in  which  8  is  contained  6 
times,  with  a  remainder  of  2. 

3.  This  2,  with  the  7,  makes  27,  in  which  8  is  contained  3  times, 
with  a  remainder  of  3. 

4.  When  3  in  the  dividend  is  reached,  it  is  found  that  8  is  not 


62 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


contained  in  it,  so  a  cipher  is  placed  under  it  and  3  considered  a 


remainder.* 

Divide  the  following: 

3.  134615379-4-2 

4.  21637298452-5-3 

5.  59368217755--4 

6.  1416823687949-f-5 

Rem 

7.  13645217--6    5 

8.  23176841-:-7     2 

9.  47896739--8     3 
10.  89765432-J-9     8 


Quotients.         Rem. 

Am.  67307689—1 

Am.  7212432817—1 

Ans.  14842054438—3 

Am.  283364737589—4 

Rem. 

11.  361745731—10  1 

12.  213764952—11  5 

13.  178961521—12  1 

14.  345678900—12  0 


Total  quotients,  21546207  and  99327785. 

15.     8092375176—3=?        19.  33376231825—  7=? 


16.  7316305416- 

17.  35192364975—5=? 

18.  13138539444 


23.  65420506756—11=? 

24.  96451487844—12=? 

25.  31463215726—  7=? 

26.  21347096543—  8=? 
Total,  21148074953—12. 


-4=?        20.  72143250072—  8=? 
21.  30608807751—  9=? 
-6=?        22.  81726954390—10=? 


Total  quotients,  13754764315  and  25359613454. 


27.  57S312908-f-3=? 

28.  483796459-^-4=? 

29.  761147355-^-5=? 

30.  123450678-4-6=  ? 
Total,  486524667—3. 


31  to  34.     Divide  874109630175  by  2,  3,  4  and  5. 

Sum,  1121774025390—4. 

*  PROOF  OF  DIVISION. — Division  may  be  proved  by  multiplying 
the  quotient  by  the  divisor  and  adding  the  remainder.  Take  the 
last  example: 

1634604702-1 


12076837617 


DIVISION  63 

35  to  38.  Divide  the  same  number  by  6,  7,  8  and  9. 

Sum,  476944738684—21. 
39  to  41.  Divide  the  same  number  by  10,  11  and  12. 

Sum,  239717944032—9. 

42.  710084973-r-  7=?  46.  31463574213--7=? 

43.  394789006-f-  8=?  47.  91678543210-=-8=? 

44.  361007839-5-  9=?  48.  76303074368--9=? 

45.  909738697--10=?  49.  21356703648-f-6=? 

Sum,  281875186—17.  Sum,  27992184199—5. 

36.  The  quotient  of  a  number  divided  by  2  is  the  £ 

(one-half)  of  it  ;  divided  by  3  it  is  the  -|  (one-third)  ;  by 

4,  the  \  (one-fourth);  hence,  to   find   the  £,  \   or  \  of  a 

number,  we  have  simply  to  divide  by  2,  3  or  4.* 

Let  it  be  requited  to  find  the  cost  of  2|  yards  of  cloth 
at  $3  a  yard. 

$3  =cost  of  1  yard. 
2 

6  —cost  of  2  yards. 
\^=cost  of  \  yard. 

$7±=cost  of  2^  yards. 
Here,  to  multiply  by  2  j,  the  3  had  to  be  divided  by  2. 

50.  -J  of  3716=1238g  f  54-  7   of  34161143764=  -  « 

61.  -I  of  1367=  -  1  55.  £   of  37897181237=  -  | 

62.  ^  of  7854=  -  1  56.  J    of  16872352168=  -  | 
53.  J  of    879=  -  1  57.  TV  of  34564185432=  -  /3 


*The  learner  will  be  particular  to  observe  that  finding  the  J  or  J 
of  a  number  is  not  dividing  by  one-half  or  one-third,  but  simply 
finding  one  part  of  something  divided  into  two  or  three  parts  which 
is  multiplying  by  one-half  or  one-third. 

•  "j  The  remainder  in  this  example  being  2,  we  write  it  as  f,  which 
indicates  that  2  is  divided  by  3,  or  that  it  is  2  parts  of  something 
divided  into  3  parts.  The  remainders  only  of  the  questions  which 
follow  will  be  given. 


64  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

58.  $14567.85X9|  is  how  much? 
1456785    cents. 


13111065 
485595 

13596660  or  $135966.60 

The  decimal  point  was  removed  before  dividing  and  replaced 
after  the  operation  was  performed. 

59.  J-  of  $21372      =?  63.    £    of  $67849132.87=? 

60.  £  of  $13744      =?  64.    \    of  $16493178.00=? 

61.  i  of  $73176.35=?  65.    \    of  $23610934.10=? 

62.  \  of  $14537.07=?  66.  TV  of  $12310985.47=? 

Total,  $25092.50.  Total,  $14579927.51. 

67.  $345.78 X 37 ^=how  much? 
34578 
37A 


17289  =  \  of  multiplicand. 
242046  =  7  times  do. 
103734     =30  times  do. 


1296675  =37-| 
or  $12966.75. 

68  and  69.  $146.82x8-*=?     $1713.14x6{  =  ? 

Sum,  $11930.62,1. 
70,  71  and  72.  $4563.28X451,  16J  and  18£. 

Sum  of  products,  $365062.39,91+ 1. 
73,  74  and  75.  $21763X14},  15£,  29|. 

Total,  $1274430.916cj. 
76,  77  and  78.  $7649.14X76J,  96J,  86-j. 

Total,  $1977090.20j  +  §. 

79  to  81.  $3146X2^,  6^,  12^.  Total,     $67639.00. 

82  to  84.  $15G7X^,  5,"    llU.  $39175.00. 

85  to  87.  $7864X6|,  71,  37^.  $400670.80. 


DIVISION. 


65 


88  to  90.  $71684.25x8|,  7|,  5'. 
91  to  93.  $89647.86  X  4,  2£,  s|. 
94  to  96.  $79943.52  X  £,  6£,  71. 

Amount,  $3327494.19. 

97  to  100.  If  a  steamboat  is  worth  $3456,  what  will  |  be 
worth?  What  f  ?  What  f  ?  What  f  ?  Total,  $6912. 

REMARK. — 2  fifths  will  cost  2  times  as  much  as  1  fifth;  3  fifths, 
3  times  as  much  as  1  fifth,  etc. 

101  to  105.  TV  of  $155367  is  how  much?  -l=?  |=? 
f=?  |=?  Total,  $97720.910f 

37.  2^0  divide  by  10,  we  point  off  one  figure  on  the  right, 
by  100,  two  figures,  by  1000,  three  figures.  Those  on  the 
left  will  be  the  quotient,  those  on  the  right  the  remainder. 
§500  divided  by  10=§50.0.  $500  divided  by  100=$5.00. 

REASON. — By  pointing  off  one  figure  we  remove  all  the  figures 
one  place  further  to  the  right,  so  that  the  tens  stand  where  the  units 
were,  and  are  units,  the  hundreds  where  the  tens  were,  and  are  tens. 

2.  It  will  be  observed  that  the  number  pointed  off  corresponds 
with  the  number  of  ciphers  in  the  divisor.     For  10  we  point  off  one 
figure;  for  100,  two;  for  1000,  three. 

31456-^-100=314.56,  or  314^ 

3.  Observe,  also,  that  a  decimal  fraction,  as  .65,  is  changed  to  a 
common  fraction  by  removing  the  point  and  writing  the  figure  1, 
with  as  many  ciphers  annexed  as  there  are  figures  in  the  decimal: 


106.  134567—  10=? 

107.  34578—  100=? 

108.  71364—1000=? 

109.  21789—  100=? 

110.  2163—  100=? 

111.  7865—1000=? 


112. 
113. 
114. 
115. 
116. 


9876—  10=? 
3453—100=? 
6543—  10=? 
2197—100=? 
1763—  10=? 


117.  7974—100=? 


Total  quotients,  14121.229  and  1954.44. 
38.    To  divide  dollars  and  cents,  the  decimal  point  is  re- 
moved   to    the    left,  which    is    the    same    as  pointing  off. 
6 


66  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

To  divide  by  10,  it   is  removed  one  figure;  by  100,  two 
figures.     $55.10  divided  by  10=$5.510.     $167.56  divided 
by  100=81.67,56.* 
Divide  the  following: 

118.  $457.87-4-     10.          122.  $473.04—1000  and    100. 

119.  $1677.45-4-  100.          123.     $15.17—     10  and    100. 

120.  $6109.88^-1000.          124.     $16.57—  100  and      10. 

121.  $14999.99-4-  100.          125.  $106.07-4-  100  and  1000. 
Total,  $218.66.  Total,  $9.85,9 

126.  Divide  the  following  sums  of  money  by  100:  $645, 
$1678.25,  $87493.57,  $16453.27,  $1998.38,  $643.24,  $2168, 
$4137.54.  Total  answer,  $1152.16,9. 

39.  It  often  happens  that  there  are  not  as  many  figures 
to  cut  off  as  there  are  ciphers  in  the  divisor.  In  such 
cases  we  prefix  ciphers  to  the  dividend  to  make  up  the 
number. 

Divide  $5.  by  100.  Ans.  .05. 

EXPLANATION. — This  is  the  same  as  removing  the  decimal  point 
two  places  to  the  left,  as  above.  The  $5  had  the  decimal  point  on 
the  right  of  the  5;  it  is  now  two  places  further  to  the  left,  and 
therefore  is  divided  by  100.  The  cipher,  in  this  case,  as  elsewhere, 
possesses  no  value. 

127.  $5-4-     10=.5  132.     $0.03--  10=? 

128.  $3^-  100=.03  133.     $0.02-4-100=? 

129.  $4-f-1000=.004  134.     $0.14-=-100=? 

130.  $50-4-1000=.05  135.     $3.16-f-100=? 

131.  $457-4-1000^.457  136.  $21.30-4-  10=? 

Total,  1.041.  Total,  2.1662. 


*NOTE. — The  value  of  each  and  all  of  the  figures  decreases  ten- 
fold for  every  figure  the  decimal  point  is  removed  to  the  left. 

The  $5  in  the  first  example  became  50  cents  and  the  10  cents  be- 
came 10  mills  or  1  cent,  making  the  answer  5  dollars,  51  cents,  not 
6  dollars,  510  cents.  The  second  answer  is  7  dollars,  67  cents,  5 
mills  and  /0  of  a  mill,  or  $1.07,5^. 


^~.  \ 


DIVISION.  67 

137.  Divide   the   following  sums   by  100:  3   cents,  33 
cents,  $3.33,  $33.33,  $333.33,  $3333.33. 

Total,  $37.03,68. 

40.  To  divide  by  20,  300,  5000,  etc.,  we  point  off  as 
many  figures  in  the  dividend  as  there  are  ciphers  in  the 
divisor,  and  divide  by  the  2,  3,  5,  etc.  The  figures 
pointed  off  will  form  part  of  the  remainder.* 

138.  Divide  317745  by  500.  5 1 00)3177 1 45 

EXPLANATION — Pointing  off   two  figures,  we  635—245 

divide  by  100;   what  is  left  we  divide  by  5.  635   ?~§ 

139.  467831-4-  20=23391^1       142.  716849-=-700=? 

140.  71 6893^-300=2389^9  3       143.  897653--900=? 

141.  417368--500=  834|«g       144.     49673---  80=? 

Total  quotients,  2641.     Total  rein.,  1573. 
The   answers   to  the  following   are  required   in  dollars, 
cents  and  mills,  omitting  the  remainders: 

$2131.51—500=?     Reduced  to  mills,  it  is  2141510. 
5 | 00)21315 | 10 

4263  ^=$4.26,3^. 

145.  $13764.75-4-50=?          149.  $16789.37--  80=? 

146.  $73968.23-4-60=?          150.  $67859.67-^-900=? 

147.  $37437.18^-90=?          151.  $54168.23^-700=? 

148.  $18964.20-4-80=?    '      152.  $78910.00-4-600=? 

Total,  $2161.11,8.  Total,  $494.16,5. 

153  to  157.  Divide  the  following  sums  by  20,  and  give 

the    answers    as    above:    $1367.25,    $3143.57,    $2345.87, 

$34.57,  $45679.44.  Total,  2628.53,3. 

*When  dividing  dollars  and  cents,  reduce  them  by  erasing  the 
decimal  point  and  annexing  ciphers  if  necessary. 
$357/27  ~500~ ? 
Reduced  to  cents  the  dividend  is  35727.     5  |  00)357  |  27 

~Kffi 


68  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

158  to  165.  Divide  $34567.25  by  10,  12,  20,  100,  30, 
50,  70  arid  90.  Total,  11132.84,6. 

166  to  176.  Divide  $367897.87  by  100,  and  the  quotient 
by  10,  20,  30,  40,  50,  60,  70,  80,  90.  Total,  $4719.74,4. 

177  to  187.  Divide  $17654.37  by  100,  and  the  quotient 
by  3,  10,  7,  40,  30,  50,  70,  90  and  80.  Total,  $298.78. 

188  to  196.  Divide  $314937  by  100,  and  multiply  the 
quotient  by  7;  then  divide  the  quotient  by  30,  60,  40, 
12,  9,  80,  90.  Total,  $22968.52,7. 


•    VI.  PERCENTAGE.* 

41.  Percentage  is  the  method  of  reckoning  by  Jiun- 
dredths.  1  per  cent,  is  the  one  hundredth  part,  2  per  cent, 
twice  that  amount,  3  per  cent,  three  times  that  amount. 

42-  The  sign  is  %.     25%  signifies  25  per  cent. 

43.  To  compute  percentage,  or,  in  other  words,  to  find 
any  rate  per  cent.,  we  first  find  one  per  cent.,  and  multi- 
ply it  by  the  given  rate. 

1.  To  find  5%  of  350. 

One  per  cent,  is  the  number  divided  by  100  or  3.50. 
5  per  cent  is  3.50x5=17.50  or  17^%  or  l7|i 

2.  Find  6%  of  3572. 

OPERATION.  35.72=1  per  cent. 

6 


214.32  or  2U^=6%. 

3.  6%  of  3146  is  how  much?      Ans.  188.76  or 

4.  5%  of  1937  is  how  much?  Ans. 

*This  rule  is  of  such  general  utility,  is  so  simple  in  its  appli- 
cation, and  so  strictly  belongs  to  the  subject  of  division,  that  I  can 
not  refrain  from  introducing  it  in  this  place. 


PERCENTAGE. 


5.  7%  of  3176  is  how  much? 

6.  9%  of  7854  is  how  much? 

7.  8%  of    396  is  how  much? 

8.  4%  of    243  is  how  much? 

Amount,  970^. 

13.  20%  of  3161=? 

14.  33%  of    798=? 

15.  55%  of    654=? 

16.  19%  of    321=? 

Amount,  1316^. 

21.  12J%  of  167=? 

22.  151%  of  364=? 

23.  371%  of  910=? 

24.  50J%  of  693=? 

Amount,  7< 


9.  4%  of  1300=? 

10.  6%  of    367=? 

11.  9%  of    463=? 

12.  8%  of  6735=? 

Amount,  654.49. 

17.  30  %  of  4541=? 

18.  23  %  of    147=? 

19.  60  %  of  7163=? 

20.  33£%  of  4371=? 

Amount,  7150^. 

25.  2%  of    $320=? 

26.  3%  of    $976=? 

27.  5%  of  $8900=? 

28.  7%  of  $6540=? 

Amount,  $938.48. 

33.  20  %  of  $1361=? 

34.  451%  of    $316=? 

35.  17   %  of  $2163=? 

36.  19J%  of  $1723=? 

Amount,  1119.67,5. 

37.  25%  of  $264.50  is  how  much? 

2.6450=1% 
J 25 

132250 

52900 


29.  7%  of  $327=? 

30.  9%  of  $100=? 

31.  12%  of  $978=? 

32.  25%  of  $179=? 

Amount,  $194.00. 


66.1250  or  $66.12,5=25%. 


38.  3% 

39.  5% 

40.  6% 

41.  7% 


of  $674.75=?  43. 

of  $198.45=?  44. 

of  $786.70=?  45. 

of    $14.13=?  46. 

42.  12%  of      $1.19=?  47. 
Amount,  $78.49,8TV 


6% 
12% 
11% 
40% 
50% 


of  $397.25=? 
of  $187.17=? 
of  $710.00=? 
of  $1678.00=? 
of  $7764.82=? 


Amount,  4678.00,5T4n. 


70  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


48.  37J%  of  $461.75=?   51.  \*\%  of  $610.18=? 

49.  %\\%  of  $198.18=?   52.  \§\%  of  $114.14=? 

50.  62%%  of  $213.07=?  *Amount,  539.60,8  r605ff. 

GENERAL  EXERCISES. 

1.  Divide  $367.22  equally  among  7  persons.     Each  will 
have  ^;  divide  by  7. 

2.  12  horses,  of  equal  value,  cost  $3456.24;   what  was 
the  cost  of  each? 

3.  If  5  men  accomplish  a  piece  of  work  in  320   days, 
how  long  will  it  take  one  man  to  do  it? 

4.  In  a  year  there  are  365  days;  how  many  weeks  will 
that  make? 

5.  A  ship  sails  84  miles  in  12  hours;  what  is  her  aver- 
age speed  per  hour? 

6.  How   many   shillings   in    13456    pence?    how   many 
pounds? 

7.  If  30  bricklayers  can  erect  the  walls  of  a  house  in 
120  days,  how  long  will  it  take  12  to  do  it? 

8.  How  long  will  it  take  a  writer  to  copy  a  speech  of 
22340  words,  if  he  writes  40  words  a  minute  ? 

9.  If  $345.72  be  divided  among  12  persons,  how  much 
will  each  receive? 

10.  In  a  pound  there  are  20  shillings;  how  many  pounds 
are  there  in  3456  shillings? 

11.  How  many  shillings  are  there  in  21345  pence?  how 
many  pounds? 

12.  How   many   bushels    of  wheat   in    134563    pounds, 
reckoning  60  pounds  to  the  bushel? 

Answers:    1600,   $52.46,   $288.02,   52},   2242f£,  7,  16, 
£88  18  9,  9,  £172  16,  300,  5581  $28.81,  1121T42,  1778  9. 

*The  Teacher  should  give  numerous  oral  exercises  on  this  rule, 
else  scholars  will  be  apt  to  err  in  pointing  the  products. 


GENERAL  EXERCISES,  71 

13.  A  and  B   ar-e   in   partnership,  in   which  A   invests 
$30  and  B  §20.     They  make  §14;  how  much  should  each 
receive  ? 

A's  share  is  30  parts. 
J3  $  share  is  20  parts. 

Together  they  have  50  parts. 

1  part  of  $14  divided  into  50  parts=28  cents;  and  20  parts= 
$5.60,  nnd  30  partsr=r$8.40,  which,  added  together— $14. 

14.  2  men  trading  horses  put  in  each  $1200  and  $800, 
and  gained  $1250;  what  was  each  man's  share? 

15.  If  20  men  do  a  piece  of  work  in  30  days,  how  long 
will  it  take  1  man  to  do  it?  how  long  15? 

16.  In  3683  oranges  how  many  dozen? 

17.  If  $3678.21    be    divided   between    7   persons,   how 
much  will  each  receive? 

18.  The  profits  of  a  speculation  in  which  was  invested, 
by  3  persons,  $300,  $900  and  $800,  are  $919.18;  how  much 
should  each  receive? 

19.  The  cost  of  muslin  is  30  cents  per  yard;  how  many 
yards  can  be  bought  for  $397  at  that  rate? 

20.  Find   the  cost   of  the   following   articles:    125   Ibs. 
sugar  at  27  cents  a  pound;  37  Ibs.  of  butter  at  37Jc;  2 
hams,  each  weighing  13  and  14  Ibs.,  at  21c;  115  Ibs.  of 
cheese  at  15J  c. 

21.  How  much  money  will  buy  37 \  Ibs.  of  tea  at  $2  a 
pound;  150  Ibs.  of  fish  at  8Jc;  57  Ibs.  of  sugar  at  31Jc; 
56J  Ibs.  of  lard  at  16  c ;  45  Ibs.  of  soap  at  7J  c;  31  Ibs.  of 
candles  at  16  c? 

22.  A  lady  bought   the   following   goods  and   paid  for 
them  out  of  $100;  how  much  had  she  left?     8J  yds.  of 
French  merino  @  $2.25;  2  pieces  brown  muslin,  33  ard 
50J  yds.,  at  35  c;  1  bonnet  at  $7.50;  1  shawl  at- $11.50. 
-   Answers :  750,  500,  600,  40,  306f £,  $23.73,7,* ,  $33.11£, 
$525.45f,  1323&  $71.12,  $367.67,2,  $137.87;T  $413.63,1. 


'72  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


VII.  LONG  DIVISION* 

THE  previous  operations  in  division  were  performed  al- 
most mentally,  the  learner  writing  only  the  quotients. 
That  method  is  preferable  when  the  divisor  is  found  in 
the  tables,  or  can  be  reduced  to  a  number  contained  in 
them,  as  500,  1200;  otherwise  the  operation  would  be  too 
difficult  and  tedious  to  perform  mentally. 

44.  Long  Division  has  then  to  be  used,  which  consists 
in  writing  the  products  and  remainders  as  well  as  the 
quotients.  The  better  to  illustrate  this  method,  an  ex- 
ample which  can  be  solved  by  short  division  is  selected. 

1.  Divide  3147  by  6. 

Divr.   Dii-d.    Quot.          EXPLANATION. — 1.  To  perform  this  operation, 
o)ol47(5^4g     we  try,  as  before,  how  often  the  divisor  is  con- 
tained in  part  of  the  dividend.     6  is  contained 
14  in  31,  5  times,  and  writing  5  in  the  quotient,  we 

12  multiply  the  divisor  by  it  and  write  the  product 

—  (30)  under  31. 

2.  We  now  subtract  the  30  from  the  31  as  we 
v/r 

would  perform  an  operation  in  subtraction.    The 

remainder  is  1.  Instead  of  supposing  this  1  to 
stand  before  the  4  in  the  dividend,  we  bring  down  the  4  to  it,  mak- 
ing 14. 

3.  6  in  14  is  contained  2  times,  which  2  we  write  in  the  quotient, 
multiply  it  upon  the  divisor  (6)  and  write  the  product  underneath. 

4.  Subtracting  this  12  from  the  14,  we  have  a  remainder  of  2,  to 
which  we  annex  7  from  the  dividend,  making  27. 

5.  6  in  27,  4  times.     Writing  the  4  in  the  quotient,  we  multiply 
it  upon  6,  making  24,  which  we  write  underneath. 

6.  Subtracting,  as  before,  we  find  a  remainder  of  3,  and  as  there 

*This,  considered  the  most  difficult  rule  in  arithmetic,  may  be 
deferred  until  the  learner  has  passed  Easy  Fractions. 


LONG  DIVISION.  73 

are  no  more  figures  in  the  dividend  to  bring  down,  we  consider  this 
the  final  remainder  and  express  it  in  fractional  form,  as  in  short 
division,  |. 

To  enable  the  learner  to  comprehend  this  part  of  divis- 
ion more  clearly,  another  example  is  introduced. 

3.  Divide  8317517  by  723. 
723)8317517(11504} ~5       1  The  product  of  the  divisor  (723)  and 

1  723*  !  I  of  the  quotient. 

2  Tno'r  2  The  remainder  of  831—723,  with  7 
™°!                                annexed. 

*^°  3  The  product  of  723  and  the  second 

4  3645  figure  of  the  quotient. 

6  3615  4  The  remainder  of  1087—723,  with  6 

6  orj-j  n  annexed. 

7  2892  5  The  Product  of  723X5)  tne  third  fig- 

ure of  the  quotient. 

8  125  6  The  remainder,  with  two  figures  (1 

and  7)  annexed.  723  was  not  contained  in  301,  so  another  figure 
was  annexed. 

7  The  product  of  723X4,  the  fifth  figure  of  the  quotient. 

8  The  remainder.     This  is  represented  in  fractional  form  in  the 
quotient. 

REMARKS. — 1.  Instead  of  using  the  whole  divisor  in  finding  a 
quotient  figure,  it  will  generally  do  to  use  only  the  first  one  or  two 
figures.  In  the  preceding  exercise,  the  first  figure  alone  (7)  was 
used,  and  in  this  way:  "7  is  contained  in  8  how  many  times?" 

2.  The  products  should  never  exceed  the  numbers  above  them. 
Number  3  should  not  exceed  number  2.     If  on  trial,  it  is  found  they 
do  so,  then  a  smaller  number  should  be  put  in  the  quotient. 

3.  For  every  figure  brought  down  from  the  dividend,  there  should 
be  one  in  the  quotient.     When  the  divisor  is  not  contained  in  the 
new  dividend,  a  cipher  should  be  placed  in  the  quotient  and  another 
figure  annexed. 

4.  The  divisor  can  not  be  contained  more  than  9  times  in  the 
smaller  dividends,  as  1087,  3645. 

*  The  learner  should  put  a  mark  under  each  figure  brought  down 
to  prevent  its  being  taken  twice. 

T 


74 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


4.       71036- 
5.       31978- 
6.     167864- 
7.  9765837- 
8.         1763- 

-  21=     3382^4 
-  43=:       743f| 
-  54=     3108ff 
-  65=150243|f 

-  76=? 

13.     3167- 
14.  71438- 

15.  67898- 
16.  78637- 
17.  1000Q- 

-  129=? 
-  320=? 
-  764=? 

-  892=? 
-7109=? 

9.         7964- 

-  87=? 

18.     7185- 

-1990=? 

0.       89737- 

-  98=? 

19.  67416- 

-  144=? 

1.       77168- 

-  19=? 

20.     3784- 

-  642=? 

2.         3167- 

-119=? 

21.  14098- 

-  671=? 

52.  *$730.45-r-126=?             25 

.  $89289.61—295=? 

53.     $164.87-i-144=?             26 

$21008.97—444=? 

24.   $1710.14-^166=?  27.  $10000.00—180=? 

Answers:  8  to  12,  inclusive,  quotients,  5116;  remain- 
ders, 211.  13  to  16,  quotients,  423  and  956.  17  to  21, 
498  and  19246.  22  to  24,  $39.54;  remainder,  6.  25  to 
27,  $405.53;  remainder,  629. 

When  there  are  ciphers  in  the  divisor,  they  may  be 
pointed  off  with  a  corresponding  number  of  figures  in  the 
dividend. 

28.  67314968-^-163000  is  how  much? 

163 1 000)67314 1 968(412}|f  -g « § 
652 


211 
163 


The  figures  pointed  off  in   the   dividend 
were  annexed  to  the  remainder,  forming  the 
484        fraction,  jff  |gg. 
326 


158 

29.  12986745—  7300=? 

30.  81098670—18000=? 

31.  513643—  2500=? 
Quot's,  6489;  Kern.,  9858, 


32.  7613412--  37100=? 

33.  4567800-;-  20900=? 

34.  5632710-^-171000=? 
Quot's,  455;  Hem.;  180222. 


*  Express  the  dollars  in  cents  before  dividing. 


LONG  DIVISION.  75 

35.  Find  the   sum,  difference,  product  and  quotient  of 
128097  and  8070;  of  1736009  and  4761 ;  and  4070391  and 
71068,  omitting  the  remainders. 

Totals,  289282688427,  1033998999,  8268611231  and 
29993710999. 

PRINCIPLES  OF  DIVISION. 

45.  If  we  divide  the  price  of  a  number  of  things  of 
equal  value  by  the  number,  we  obtain  the  price  of  one. 

46.  The  quotient  will  usually  be  in  the  same  name  with 
the  dividend   or  number  to  be  divided.     If  the   dividend 
be  dollars,  the  quotient  will  be  dollars ;  if  it  be  rods,  the 
quotient  will  be  rods. 

36.  If  75  barrels  of  flour  cost  $450,  what  was  the  price 
per  barrel? 

37.  If  125  horses  cost   $25000,  what  was  the  cost  of 
each? 

38.  If  $167809   be  divided   among  7614  persons,  how 
much  should  each  receive? 

39.  How  much  tax  should  each  of  16785  persons   pay 
of  a  levy  of  $71683? 

40.  In  303,656,837  Ibs.  of  cotton,  how  many  bales,  sup- 
posing each  bale  to  weigh  320  Ibs.? 

41.  If  $79640  be  divided  among  274  persons,  how  much 
will  each  get? 

42.  The   earth   moves   round   the    sun   at   the  rate   of 
66600   miles   an  hour;    at  what  rate  does   it    move   per 
minute? 

43."  If  357  yards  of  broadcloth  cost  $1035.30,  what  was 
the  cost  per  yard?  and  what  would  be  the  cost  of  50 
yards  at  the  same  rate? 

Answers   arranged  promiscuously:    1110,    948, 
$290.65,  $6,  $22.30,9,  $145,  $4.27,7,  $200,  $2.90. 


76  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


EXERCISES  IN  MULTIPLICATION  AND  DIVISION.* 

1.  If  23  yds.  of  inuslin  cost  $3.45,  what  will  one  yard 
cost? 

2.  If  117  men  can  do  a  piece  of  work  in  48  days,  how 
long  will  it  take  3  times  that  number  to  do  it? 

3.  How  many  men  can  do  a  piece  of  work   in   5  days, 
that  took  10  men  25  days? 

4.  If  a  case  holds  29  pieces  of  muslin,  how  many  will 
it  take  to  hold  7250  pieces? 

5.  If  15   men   can    do   a   certain   piece    of  work   in   75 
days,  how  long  will  it  take  1  man  to  do  it? 

6.  If  7    dozen   silver   spoons   cost  $35.35,  what  will   3 
dozen  cost? 

Find  the  cost  of  one  dozen,  then  the  cost  of  3. 

7.  If  f  of  a  ship  cost  $14602,  what  will  the  ^,  or  the 
whole  ship  cost? 

8.  If  |  of  a  piece  of  property  cost  $6377,  what  will  -J 
of  it  cost? 

9.  In   a   cord   of  wood   there   are   128   feet,  how  many 
cords  are  in  a  pile  measuring  4  feet  wide,  8  feet  deep  and 
100  feet  long? 

10.  In  an  acre  there  are  4840  square  yards:  how  many 
are  there  in  -J-  of  an  acre  ? 

11.  A  field  contains,  12  acres,  and   is  660  yards  long; 
how  many  yards  is  it  in  breadth? 

12.  A   tract   of  5    acres  is  220  yards  long;  how  much 
should  be  cut  off  the  breadth  to  leave  1  acre? 

13.  -J-  of  a  dozen  books  cost  $7.50;  what  was  the  cost 
per  dozen? 

14.  §  of  a  dozen  cost  22  cents;  what  was  the  cost  per 
dozen? 

*For  answers,  see  next  page. 


PROPERTIES  OF  NUMBERS.  77 

15.  1^  dozen  cost  $1.20;  what  was  the  cost  per  dozen? 

16.  T7~   of  a  hundred  cost  $28;  what  was  the  cost  per 
hundred? 

17.  23jy  of  a  hundred  cost  |15.75;  what  was  the  cost  per 
hundred  ? 

18.  1 1  of  a  hundred  cost  $3.15 ;  what  was  the  price  per 
hundred? 

Answers:  51107,  3644,  25,  16,  50,  250,  22,  15.15,  605, 
60,  88,  80,  33,  100,  4.20,  105,  1125,  15,  88. 

47.  METHOD  OF  PROOF. — Division   and   Multiplication 
beiDg  converse  operations,  the  one  is  proved  by  the  other. 

DIVISION.  PROOF. 

38)3715(97  97=quotient. 

342  3S=divisor. 

~295  776 

266  291 


29  rem.      3686+tfie  rem.,  (29°)=37lb=dividend. 

MULTIPLICATION.  PROOF. 

multiplier,  product,  multiplicand. 

465  25)11625(465 

_25  100 

2325  162 

930  150 


11625  125 

125 


VIII.  PROPERTIES  OF  NUMBERS. 

48.  An  Integer  is  any  number  considered  as  a  whole, 
as  3,  7.  58,  129. 

49.  A  Fraction  is  a  part  of  any  thing  or  number  of 
things  considered  as  a  whole;  ^,  -?-,  -f-fc. 

50.  Numbers  are  divided  into  Odd  and  Even. 


78  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

An  Odd  number  can  not  be  divided  into  two  equal  parts 
without  a  remainder,  as  1,  5,  57. 

An  Even  number  can  be  divided  into  two  equal  parts 
without  a  remainder,  as  4,  10,  68. 

51.  Numbers   are   either  Prime  or   Composite,  Abstract 
or  Concrete. 

52.  A  Prime   number   is   an    original    number,   or   one 
which    can    not   be    produced    by    multiplying    two    other 
numbers  together,  as  1,  7,  31. 

53.  A  Composite   number   is    one  which   may   be   com- 
posed  of  two    other   numbers    multiplied    together,    as   8, 
which   is  composed   of  2  and  4  multiplied  together;   and 
27,  which  is  composed  of  9  and  3  multiplied  together.* 

Exercise. — Write  out  50  prime  and  50  composite  numbers. 

54.  An  Abstract   number   is   an   unapplied  number,   or 
one  which  conveys  the  idea  of  number  exclusively,  as  4, 
15,  47. 

55.  A  Concrete  number   is   an   applied  number,  or  one 
which  conveys  the  idea  of  something  else  besides  number. 
The  above  numbers  become  concrete  when  applied  as  fol- 
lows :    4    mills,    15    dollars,    47    pounds,    and    the    names, 
mills,  dollars  and  pounds,  are  called  denominations. 

56.  A  Multiple   is    a    number  which    contains   another 
number  a  certain  number  of  times   without  a  remainder. 
12  is  a  multiple  of  3  as  well  as  of  2,  4  and  6. 

57.  A  Common  Multiple  is  one  which  contains  two  or 
more    numbers  a  certain    number   of  times   without  a  re- 
mainder.    12  is  a  common  multiple  of  2,  3,  and  4. 

58.  The  Least   Common  Multiple   is   the   least   number 


*NOTE. — Since  all  even  numbers  are  divisible  by  2,  an  even 
number  can  not  be  a  prime  number,  nor  can  any  number  ending 
with  5;  it  follows,  therefore,  that  every  prime  number,  except  2 
arid  5,  ends  with  1,  3,  7  or  9. 


PROPERTIES  OF  NUMBERS.  79 

which  will  contain  two  or  more  numbers  without  a  re- 
mainder. 6  is  the  least  common  multiple  of  3  and  2,  and 
18  of  3  and  6,  and  24  of  2,  3  and  8. 

59.  An  Aliquot  is  a  number  which  will  divide  another 
without  a  remainder.     The  parts  of  which    a  multiple  is 
composed  are  called  aliquot  parts  of  that  number.     1,  2, 
3,  4  and  6  are  aliquot  parts  of  12. 

60.  Complement:    The    number  required    to  be   added 
to  another  to  make  it  equal  to  a  larger.     It  is  usually  ap- 
plied to  100,  1000  or  some  other  power  of  10.     Taking 
87  as  a  part  of  100,  the  complement   is    13,  or  50  as  a 
part  of  60,  the  complement  is  10. 

Exercise. — Taking  35  as  a  part  of  60,  70  as  a  part  of 
100,  18  as  a  part  of  20,  73  as  a  part  of  80,  required  the 
complements. 

61.  Even    numbers    are    divisible    by   2  without   a   re- 
mainder. 

62.  If  the  two  right-hand  figures  of  a  number  are  di- 
visible  by  4  without  a  remainder,  the  whole  nun^ber  is 
also  divisible  by  4. 

63.  Numbers    ending  with    5    or  0  are    divisible  by  5 
without  a  remainder. 

64.  If  the  three  right-hand  figures  of  a  number  are  di- 
visible by  8  without  a  remainder,  the  whole    number  is 
divisible  by  8. 

65.  If  the  sum  of  the  figures  of  any  number  is  divisible 
by  3  or  9  without  a  remainder,  the  whole  number  will  be 
divisible  by  3  or  9. 

MULTIPLICATION  BY  ALIQUOTS. 

66.  To  multiply  by  2-i,  it  will  shorten  the  operation  if 
we  multiply  by  10,  which  is  4  times  too  much,  and  then 
divide  by  4.     In   the  same  way  we  can  multiply  by  any 
other  aliquot  of  10,  or  by  aliquots  of  100,  1000,  etc. 


80  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

To  multiply  176  by  12|. 

8^17600  EXPLANATION:  —  176  being  multiplied  by  100  is 

.  -  8  times  more  than  the  sum  required,  so  we  divide 

2200  Ans.    by  8> 

To  multiply  379  by  250. 
4)379000  379  being  muitiplied  by  1000  is  4  times  too 

94750  Ans.     much,  so  we  divide  by  4. 
To  multiply  $49.75  by  125. 

8)4975  000  The  $49.75  are  considered  as  cents  and  mul- 

/>oi  o-re  tiplied  as  the  preceding. 

621  8*5  cents.      *!  ..          .     ..,. 

A/irt-m  *r     A  REMARK.  —  It  will  lessen  the  work  still  more 

or  $6218.75  AM.  .     .     ,  *  14.  ..  , 

to  simply  assume  the  number  to  be  multiplied 

by  10,  100  or  1000. 

ALIQUOT  PARTS  OF   10,   100,   1000. 

To  be  committed  to  memory. 

ALIQUOTS  OF  10.  ALIQUOTS  OF  100.  ALIQUOTS  OF  1000. 

5  =4  50  =4  142=4  333£=i 

3i^i  331=4  12£=£  250=1 

2i=3  25  =i  10  =T^  166§^  I 

2  =1  20  =i  8^=^  125  =^ 


The  pupil  can  prove  the  accuracy  of  his  calculations  by 
multiplying  in  the  ordinary  way. 

4.  140X  12i=?  13.  949  x333i=? 

5.  3767  X     8|=?  14.  179  X     2-i=? 

6.  9987X  25°^?  15.  769  X     3i=? 

7.  9174X125  =?  16.     12-'  X   19  =? 

8.  3689X  33^  =  ?  17.  125^X787  =? 

9.  9210X   16'j=?  18.  250   X125  =? 

10.  7897X166^=?  19.     16*  X  48  =? 

11.  8997X   50  =?  20.     83^x756  =? 

12.  786X  14$=?  21.  197   X   12£=? 


PROPERTIES  OF  NUMBERS.  81 

23.  675  yards  @  37^  cents. 

OPERATION.     675  at  &  dollar—  $675.00 

at  25  c=J   "168/75" 
at  12^  c^J-       84.374 

^w.  253^121 

24.  715X62J  cents.  31.  9876x$2.18f 

25.  947XS7A  32.     719x?3.62£ 

26.  194xl8f  33.     965x$4.37J 

27.  567X31}  34.     758x$l-25~ 

28.  619X374  35.     197x$2.87.V 

29.  1060X324  36.     879x$3.95~ 

30.  197X75"  37.     179XW.32J 

38.  To  find  the  cost  where  there  are  fractions  in  both 
factors:  18|  Ibs.  @  12i  cents. 

OPERATION.     18|  Ibs.  @  $1=$18.75 

at  12in=^  or    $2.34|  Arts. 

39.  37J  Ibs.  @  18|  cents. 

OPERATION.     37^  @  $1=837.50 

at  12^=-^  4.687 
at    6^=4  2.343 

Am.  $7.030 

40.  176^  doz.  @  18|  cents.       41.  164J  Ibs.  @  $1.62J 
73l|  doz.  @  12J  374J  Ibs.  @    2.25 

doz.  @  37J  693J  Ibs.  @    0.16f 

Amount,  $197.06.  Amount,  $1224.93.    ' 


E.  —  The  multiplier  in  the  25th  Ex.  wants  only  12J  cents,  or  J, 
of  being  a  dollar;  so  we  find  the  cost  of  947  at  a  dollar  and  take 
off  I  of  it.  For  32J  take  30,  and  2J  as  J-  of  10. 


ill 

/rf 


82  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


IX.  EASY  FRACTIONS.* 

67.  A  FRACTION  is  a  part  or  number  of  parts  of  any 
thing  considered  as  a  whole.     Fractions  are  of  two  kinds, 
common  and  decimal.     A  common  fraction  is  written  with 
two  numbers,  called  terms,  having  a  line  between  them,  as 
•j^;  a  decimal  fraction  with   one   number,  having  a  period 
at  the  left,  as  .5  (five-tenths). 

68.  A   common   fraction   indicates  division,  the  upper 
number  being  the  dividend  and  the  lower  the  divisor.     In 
treating  of  fractions,  the  dividend  is  called  the  numerator 
and  the  divisor  the  denominator. 

The  denominator  indicates  the  number  of  parts  into 
which  the  whole  is  divided,  and  the  numerator  the  num- 
ber of  such  parts  under  consideration. 

69.  VALUE  OF  A   FRACTION. — The   lowest  value  of  a 
fraction  is  expressed  by  the  figure  1  for  a  numerator,  and 
the  highest  value  a  number  as  great  as  the  denominator 
less  l.f     ^  represents  the  lowest  value  of  fractions  of  the 
denomination   of  ninths,   while  .  |   represents   the  highest 
value  of  that  denomination.  J 

*This  chapter  is  introduced  for  the  benefit  of  that  large  class  of 
pcholars  who  leave  school  before  completing  the  study  of  Arith- 
metic. The  subject  of  fractions  is  treated  of  at  length  in  the  latter 
part  of  this  book. 

|This  does  not  apply  to  improper  fractions,  which,  as  the  name 
indicates,  are  not  strictly  fractions. 

£1.  Since  this  is  the  case,  it  is  evident  that  fractions  decrease  in 
value  as  their  denominators  increase,  the  numerators  remaining  the 
same.  1  is  less  than  ^,  1  than  1,  £  than  1. 

2.  It  is  also  evident  that  the  value  of  a  fraction  depends  on  the 


EASY  FRACTIONS.  83 

When  a  number  is  divided  into  two  parts,  each  part  is 
called  a  half  ;  into  3  parts,  each  part  is  called  a  third; 
into  4  parts,  each  part  is  called  a  fourth;  into  5,  a  fifth; 
into  12,  a  twelfth;  into  18,  an  eighteenth;  into  25,  a  twen- 
ty-fifth; into  100,  a  hundredth;  into  476,  a  four  hundred 
and  seventy-sixth  part. 

ORAL  EXERCISES. 

1.  When  a  number  is    divided   into   10   parts,  what  is 
each   part  called?     Into  11?     Into   20?     Into   33?     Into 
45?     Into  97?     Into  62? 

2.  When  divided  into  31,  what?     Into  69?     Into  103? 
Into  364?     Into  155?     Into  1000?     Into  3144? 

3.  Which  is  the  greater  fraction,  ^  or  J  ?     -|  or  J?     £ 

o'i?    TY°rT's?    2'a°r  ,',?    ,'s  or  &? 

4.  Which  is  greater,  f  or  |?  Ans.  |. 

REASON.  —  Because  it  will  take  less  to  make  it  a  whole  number. 
The  first  fraction  requires  J  to  make  it  a  whole  number,  while  this 
one  requires  only  J. 

5.  Which  is  the  greater,  f  or  f  ?     |orf?     |or|?     f 
or  I?     f  or  I?     ?orTy     if  or  ji?     f  or  ?? 

6.  Which  is  the  greater,  if  or  ff  ?     ^f  or  f|?     jf  or 
H?     i*or£f?     IforAf?     14  or  if? 

Since  the  value  of  a  fraction  depends  upon  the  relation 
of  the  numerator  to  the  denominator,  [note  2,  page  82,] 
both  terms  may  be  multiplied  or  divided  by  the  same 
number  without  altering  its  value. 

_4  ?^-2_1 

~  ~~ 


relation  of  the  numerator  to  the  denominator,  or,  in  other  words, 
the  number  of  times  the  numerator  is  contained  in  the  denomina- 
tor. |  is  equal  to  |,  because  the  numerator  3  is  contained  in  its 
denominator,  6,  the  same  number  of  times  that  the  numerator  4  ia 
contained  in  the  denominator  8. 


£4  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Now,  |  and  i  possess  the  same  value  as  f  ,  because 
their  respective  numerators  are  contained  the  same  num- 
ber of  times  in  their  denominators. 

EXERCISES  FOR  THE  SLATE. 

5.  Change  |  to  twentieths. 

3X&  15  EXPLANATION.  —  By  multiplying  the  4  by  5,  we  change 
4X&  ^0  the  denominator  to  twentieths;  arid  by  multiplying  the 
numerator  by  the  same  number  we  preserve  the  same  value. 

1.  Change  -|    to  8ths;    £  to    12ths;    4    to   20ths;    f   to 
Uths;  I  to  12ths;   f  to  18ths;  £  to  BOths. 

2.  Change  ^  to  20ths;  f  to  IGths;  f  to  27ths;  j|  to 
52ds;  |  to  25ths;  fg  to  150ths. 

3.  Change  |  to  32ds;   £g  to  40ths;  Jf   to  72ds;  fg  to 
104ths. 

4.  Change  £§  to  SOths;  if  to  52ds;  ||  to  128ths;  T40 
to  50ths. 

5.  Change  T^  to  6ths  ;    ^  to  5ths;  T82  to  4ths;  j|  to 
8ths;  if  to  7ths;  j§  to  8ths. 

6.  Change  |  to  halves;  T6a  to  5ths;  T4^  to  4ths;  ||  to 
8ths;  fg  to  3ds. 

The  fractions  in  last  exercise,  (6th,)  when  changed  as 
required,'  would  be  reduced  to  their  lowest  terms;  that  is, 
expressed  in  their  simplest  form. 

70.  To  reduce  a  fraction  to  its  lowest  terms  is,  there- 
fore, to  divide  the  numerator  and  denominator  by  such  a 
number  or  numbers  as  will  do  so  without  a  remainder. 
When  the  terms  can  not  be  divided  exactly  by  any  num- 
ber greater  than  1,  the  fraction  is  in  its  simplest  form. 

Reduce  T62  to  its  lowest  terms. 


Ty,  T?,      ,  T|g,  ^  -fa  to  their  lowest  terms, 
Answers:  £,  £,  T'r,  &,  i,  4,  ^. 


EASY  FRACTIONS.  85 

When  a  single  number  will  not  reduce  the  fraction,  other 
numbers  may  be  used,  as  below. 
2.  Reduce  -§-f  f-g-  to  its  lowest  terms. 


3.  Reduce  to  their  lowest  terms,  -££%,  -±f^,  -J|f,  and 

4.  Reduce  to  their  lowest  terms,  -£££$•>  TT^  sWb^ 

T5~3T' 

5.  Reduce  to  their  lowest  terms,  ^7e47%,  ^Irolb  anc^ 
Answers  :  £ff,  ffr,  A.  H-  A>  A.  Mi.  «f,  T¥&>  #7, 

TTHF5  ITT'  "8"' 

Fractions  may  be  Proper,  Improper,  Simple,  Compound 
or  Complex.  We  shall  treat  of  only  the  three  former  at 
present. 

A  proper  fraction  is  one  whose  numerator  is  less  than 
its  denominator,  as  ^.  An  improper  fraction  is  one  whose 
numerator  is  equal  to  or  greater  than  its  denominator, 

nc    5       P 

as  TJ,  TJ. 

71.  A  simple  fraction  is  a  single  fraction,  and  may  be 
proper  or  improper,  as  |,  |. 

72.  When  a  whole  number  and  fraction  appear  together, 
they  are  called  a  mixed  number,  as  5|. 

73.  Improper  fractions  may  be  changed  to  whole  or  mixed 
numbers  by  dividing  the  numerator  by  the  denominator.* 

To  change  -1/-  to  a  mixed  number. 

5)13          EXPLANATION.  —  There  are  5  fifths  in  one  whole  num- 
~T3    ber;  in  13  fifths  there  are  as  many  Is  as  the  number  of 
5    times  5  is  contained  in  13,  which  is  two  times,  with  3 
fifths  over,  making  2^. 

*This  is  simply  acting  on  the  principle  that  the  numerator  is  the 
dividend  and  the  denominator  the  divisor. 


86  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


1.  Change  |,  g,  |,  Jg&,  2.^-L,  ££*.  to  whole  or  mixed  num- 
bers. 

2.  Change  the  following  :  -*/,  ^_,  *«-,  -2T2-,  ^-,  -^-,  1§£. 

3.  Change  i-f  o,  ^-,  -^  ff,  44,1,  ^,  AJa- 
Answers:  lj,  If,  3,  H,  301,  691,  53,.  8£,  811,  7,  58^, 

56f,  184,  13|,  3^,  81,  Hi,  4i,  91,  18f  . 

74.  To  change  whole  or  mixed  numbers  to  improper  frac- 
tions is  an  operation  the  reverse  of  the  last,  which,  scarcely 
needs  explanation. 

4.  Change  94  to  an  improper  fraction. 

9i          EXPLANATION.  —  In  1  whole  number  there  are  5  fifths; 
5        in  9  there  are  9  times  5  or  5  times  9  fifths,  to  which  we 
"JjJ        add  4  fifths,  and  we  have4^. 

5 

5.  *  Change  the  following  mixed   numbers  to  improper 
fractions:  3|,  9|,  8?,  5|,  41f,  97J-,  16|. 

6.  Change   the  following:   7|,  9}J,  4|,  7  If,  18|,  16§, 
124. 

7.  Change  the  following:  21  to  fifths;  16|  to  eighths; 
121  to  fourths;  16§  to  twelfths;   8£  to   twelfths;  131  to 
sixteenths. 

75.  To  multiply  a  fraction  by  a  whole  number  is  simply 
to  multiply  the   numerator  without  altering   the   denomi- 
nator, or  to  divide   the  denominator  without  altering  the 
numerator. 

To  multiply  T72  by  6. 


REASON.  —  Assuming  that  7  is  a  whole  number,  multiplying  it  by 
6  gives  42;  but  since  it  is  not  a  whole  number,  but  twelfths,  the  42 
is  =8-  or  3. 


*The  learner  should  prove  the  accuracy  of  his  work  by  last  ar- 
ticle. 


EASY  FRACTIONS.  87 

2-  6)/3(.J=3J. 

By  decreasing  the  denominator,  the  fraction  is  in- 
creased (as  it  takes  fewer  of  the  small  parts  to  make  a 
whole  number)  ;  hence,  the  7  represents  halves  instead  of 
twelfths.  £=3J. 

1.  |  X7=?  5.  J>uxl2=?  9.  i*xll=? 

2.  I  X9=?  6.  f|x  6—?  10.  f|Xl2=±? 

3.  5x8=?  7.  fix  5-=?  11.  Jfx  8=? 

4.  T*BX4=?  8.  i|XH=?  12.  £§X21=? 
Answers:  l-£,  6|,  5±,  5f,  10|,  9&,  7|,  4ft,  4j,  6,  1||, 

12|,  6. 

76.  To  multiply  a  whole  number  by  a  fraction,  we  mul- 
tiply the  numerator  without  altering  the  denominator. 

13.  Multiply  25  by  |. 

25X3  fourths=75  fourths,  or  7?5,  which,  changed  to  a  mixed 
number,  [Art.  73]=18j. 

14.  35X|^-?       18.  134X21o-=?        22.  16XT3o=? 

15.  21xf=?       19.  215X1—  ?       23.  21X|=? 

16.  18X^=?       20.112X|=?       24. 

17.  116XA=?      21.     36X|=?       25. 

Answers:  28,  6/ff,  4|,  174,  95|,  18,  3|,  931,  12,  i|,  20, 
34|,  30|, 

77.  To  multiply  a  mixed  number  by  a  whole  number. 
Multiply  7|  by  9. 

•  4  EXPLANATION.  —  3  fourths  multiplied  by  9=27  fourths, 

9          or  6|;  and  the  7  multiplied  by  nme=63,  plus  the  6—69, 
69^        making  the  product  69|. 

Or  thus:  7| 


The  Teacher  will  find  it  important  to  require  the  learner  to  preserve 
the  process,  as  he  will  be  apt  to  adopt  clumsy  methods  of  solution. 


88  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

2i?.  1SJX  5=?  30.  29JX  8=?  34.  S3  i  X  7=? 

27.  :>:\\  8=?  31.  IG-x  9=  ?  35.  12^  X  8=? 

28.  12-.i\12^=?  :>-•  8t|xl2=?  36.  5.,-gX  9—? 
21).  :^\   9=?  33.  62JX  9=?  37.  187  £  XH=? 

Answers:  3!,  93J,  3tf  62£,  45  ,V  96 

•:  -.  i;>»>.  1050,  i;>o.  ;u>o." 

78.  ^'  multiply  a  whole  number  by  a  jni.ird  number. 
38.  Multiply  29  by  8|. 

29 

S;  FMM.  vNvriox.  —  Multiplying  -9  by  i!  thirds,   we   hare 

TIT      58  thirxls.  or  l^J,  which  we  write  in  the  first  line.     Then 
*Z*     29X8=232,  which,  added  to  1%=251  j. 

Or  thus  :  29X-V=13d=-51  ] 

15X  3]=?    42.  12\12i^?     4.ri.  14xl7f=zT 

40    27X   6|=?    43.47X37^=?     46.  29x18  1  =? 

41.  ?     44.  93X16J^?     47.  83x  6T" 

Ai>  '  :  .  L-2J.  543|, 

522^  150.  17t52^." 

To  multiply  a  fraction  by  a  fraction. 

4$.  Multiply  j  by  j. 

Assuming  the  numerator  5  to  be  ft  whole  number,  jx^-^^1-5: 
but  5  is  not  ft  whole  number,  but  5  sixths;  hence  y  is  6  timt*  too 
MK&  tj*  dirided  by  6z=||T  or  |  [Note  lr  p*ge  82.] 

79.  Hence,   to  multiply   &  fraction  by  a  fraction,  we 
iHufcyrfjr  the  tttraierafer*  together  for  a  new  numerator,  and 
the  denominators  for  a  new  denominator. 

?  which,  reduced  to  its  lowest 


*It  will  be  obserred  that  to  mnltiplj  by  ft  fraction  does  not  m- 
II  Mini  Ike  •ultipli^^  as  in  whole  numbers;  bat,  on  the  contrary, 
Aerates  it,  the  f  being  less  ihan  J. 

To  account  for  this,  U  Ls  only  necessary  to  remember  that  ft  whole 


EASY  FRACTIONS. 
49.    -|  X  2=?  52.  £X£=? 

5J.   !;!xW?  54-  *Xj=?  57.  g/.  ?=? 

Answers:  f,  £,  J,  f,  T^,  3$,  J,  £f,  |,  £ ,', 

80.  ^  multiply  a  mixed  number  by  a  fraction  or  a  miw.d 
number. 

58.  Multiply  15|  by  TV 

15|=^»,  which,  multiplied  by  f0=*4«J-  or  14470. 

59.  Multiply    8}  by  Ifif. 
8|=3/t  an(1  lf;3=^-     544X^ 

60.  12^/lf^  -='!      W>.  14'|X      i90=?      66. 

61.  8JX2%=?      64.  23^X      i=?      67. 

62.  37^X52*=?      65.       |Xl4^^?     68. 

Answers:   126J,  35 »,  231 J,  6T3^,  ^,  208^,  245g,  29f, 
1978f 

81.  To  divide  a  wlwle  number  by  a  fraction  or  a  mixed 
number. 

1.  Divide  315  by  |,  or,  in  other  words,  find  how  often 
|  is  contained  in  315. 

.SOLUTION. — Before  we  can  measure   315   by  fourths,   we  must 
change  it  to  fourths.     In   1   there  are  4  fourths;   in  315  there  are 
315  times  4  or  1260  fourths,  which,  divided  by  3^420.     Hence,  \  is 
t  contained  in  315  420  times. 

OPERATION.    315    or 
4 

3)1260 
430 

t  number  is  reduced  to  the  denomination  of  a  fraction  by  being  mul- 
tiplied by  it.  6Xf =18  fifths  or  3|.  Much  more  is  a  fraction  re- 
duced in  value  if  multiplied  by  a  fraction.  From  this  we  readily 

«  infer, 

2.  That  to  divide  by  a  fraction  increases  the  dividend. 

8 


90 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


136-5-?=? 


3.  27-f-J=?    5.  684-f-T»0=?    7. 

Answers:  32|,  365|,  24341,  19740,  2280,  158f,  24§. 

8.  Divide  25  °by  5 J. 

OPERATION.     25x2  halves—5^  and  5|X2— ty.     50-f- 

82.  Hence,  to  divide  by  a  fraction,  we  multiply  by  the 
denominator  and  divide  by  the  numerator,  or   invert  the 
divisor  and  proceed  as  in  multiplication. 

9.  157-4-  3|=?        12.  345-4-  6J=?  15.  195-=-16f=? 

10.  22-5-12£=?       13.  39-4-15I-?  16.  39-5-12f=? 

11.  16-5-16|=?       14.  79-4-37^?  17.  87-4-3l|=? 
To  divide  one  fraction  by  another. 

18.  Divide  f  by  |.     OPERATION.  f  Xf— ig— T9o 
EXPLANATION. — By  inverting  the  divisor,  we  obtain  ^g,  the  terms 

of  which,  being  divided  by  2,  give  T90. 

19.  9  _!_!_-?          22. 

20.  /,-*-£=?         23. 

21.  2£_j_6=?         24. 

Answers :  1|,  ff ,  f|,  38|f,  36|, 

83.  To  divide  when  either  divisor  or  dividend  is  a  mixed 
number  and  the  other  term  a  whole  number,  both  terms  may 
be  reduced  to  the  same  denomination.     [Art.  81.] 

28.  Divide  3457]  by  13. 

EXPLANATION. — The  dividend  containing  the 
fraction  of  ^,  both  terms  are  reduced  to  fourths, 
and  division  performed  as  in  whole  numbers. 
The  result  shows  that  the  divisor  is  contained 
in  the  dividend  265  times,  with  a  remainder  of 
49  fourths  [Art.  46],  or  265||  times. 


-4-=?         25.       -f-l= 


-4-  §  =? 


. 

26. 
27. 


-4-6|=? 

f,  f 


3457J 
4 

52)13829(265 
104 


342 
312 


309 
260 

49 


EASY  FRACTIONS.  91 

The  same  by  short  division. 

13)3457  1          EXPLANATION.  —  13  is  contained  in  8457,  265  times, 

ope  4  9     with  a  remainder  of  12,  which,  reduced  to  fourths, 

52     including  the  |-  in  the  dividend,  is  49  fourths.     13 

not  being  contained  in  this  an  even  number  of  times,  the  denomi- 

nator is  increased  13  times,  (which  is  the  same  as  to  decrease  the 

numerator,)  which  gives  the  same  fraction  as  by  long  division,  4|0 

29.  1398J—  56  =?         35.  1255  |  —350=? 

30.  256J--  7  =?         36.  796  J  —421=? 

31.  1939  -~  %=?         37.  467  f  —  12=? 

32.  7961?-f-300  =?         38.  214  1  —  9=? 

33.  9219  ~  6J=?         39.  713^—  8=? 

34.  1391  -v-  56£=?  40.     391Jt—     6=? 
Answers:    24}g|,    24^,  1475&,  36T\,  232^,   26ffg, 


84.  To  subtract  a  fraction  from  another  of  the  same  de- 
nomination is  simply  to  subtract  the  less  numerator  from 
the  greater. 

I.  FromT7a  take  T30. 

T70—  T30=T40    °r  I 

2-  jf-H=?  5.  T»0-TV=?  8.  T»g-^=? 

337    '25  _  ?  (\  1:2  _  9  _  9  Q  1  (  >    3  _  9 

'  42  -  42  -  '  °'  3T  -  51  —  *'  3^  -  39  -  r 

416    5  _  ?  7  16    7  _  9  10  11  _  6  _  9 

•  55  -  52  -  '  f'  45  -  45  -  f  1U'  T7   IT  -  ' 

Answers:  f,  4,  J,  T\,  J,  J,  |,  |,  T6S,  T\. 

85.  ^>  subtract  a  fraction  or  mixed  nnmber  from  a  whole 
number. 

II.  From  9  take  3|. 

The  following  formula  will  render  the  operation  simple  : 

Whole  number.    Fourths.        EXPLANATION.  —  Arranging  the  less  under 

the  greater,  we  find  we  can  not  take  3  fourths 

_  from  0  fourths;  so  a  whole  number  or  1  is 

5  1  added  to  both  terms.    In  1  there  are  4  fourths, 

QT  5i  from  which  take  3  fourths,  and  we  have  a 

remainder  of  1  fourth.     To  the  3  add  1  and 


92  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

•we  have  4,  which,  subtracted  from  9,  leaves  5,  giving  for  the  an- 
swer 5£. 

12.  13—  4J=?      15.  11—  2  J=?      18.  52—  2?4=? 

13.  15—  5£=?      16.     7—    f  =?      19.  13—  12J=? 

14.  29—121=?      17.  14—  1T32=?      20.  89—  75J=? 

A  n  <JWAr«a  •     5      £1      O5      J.1      94.6      Q2      Q3      1£1      1Q3      1Q3 

Answers.   g,  og,  Jg,  *^,  £*^,  o3,  p|,  10^,  io?,  lo^. 

86.  j?b  subtract  one  fraction  from  another  of  a  different 
denomination,  it  will  be  necessary  first  to  reduce  both  to 
the  same  or  a  common  denominator. 

21.  From  |  take  f 

By  Art.  69,  |  can  be  changed  to  56ths  by  multiplying 
both  terms  by  7,  and  f  can  be  changed  to  56ths  by  mul- 
tiplying both  terms  by  8,  giving  ||  and  |-|,  the  difference 
between  which  is  ^,  the  answer. 

It  will  be  observed  that  the  multipliers  used  in  this  case 
were  the  two  denominators,  7  and  8,  which,  multiplied  to- 
gether, give  a  common  denominator,  and  multiplied  into  the 
numerators  of  each  other  give  the  new  numerators. 
OPERATION.     |—  «=ff-4|=5iff 

22.  From  f  take  i  .      25.     |—  4=?      28. 

23.  From  f  take  \  .      26.  6J—  1=?      29. 

24.  From  f  take  T«3.      27.  l|—  J=  ?      30.  T3_—  J_^? 
Answers:   25¥,  5|,  |f,  |,  T%,  J,  f|,  |,  ^,  |. 

87«  ^  «-c?cZ  fractions  of  the  same  denomination,  the  nu- 
merators only  are  added,  and  the  sum  reduced  to  a  mixed 
number  or  its  lowest  terms. 

1.  Add  I+I+I+I- 

3  EXPLANATION.  —  Here  the  four  numerators  are  added 

6  together,  making  21  eighths,  which,  reduced  to  a  mixed 

5  number,  are  equal  to  2£. 


\ 


EASY  FRACTIONS.  93 


2- 

3       31    4    1718 
•    9~r  y  ~r$  i  3 

4. 

5- 

6- 


88.  ^>  add  fractions  of  different  denominations,  they 
should  first  be  reduced  to  a  common  denominator,  as  iu 
subtraction. 

7.  |+!-?         |+|=|+|=Vt=lf  or  If 

When  three  or  more  fractions  of  different  denomina- 
tions are  to  be  added  together,  they  may  be  reduced  to  a 
common  denominator  by  multiplying  all  the  denominators 
together,  as  above,  and  then  by  multiplying  each  numer- 
ator by  all  the  denominators  except  its  own.* 

8.  Find  the  sum  of  i+f  +  §• 

2X^X6—     48=:  Common  denominator. 
1  X  4  X  6==24=sJ'Wfcl  numerator. 
3x2X^—36—  Second  numerator. 
5x2x4—  40—  ^/ureZ  numerator. 

Wfa^Stim  of  numerators. 
Hence,  ^=2^=2^. 

The  J  in  the  example  was  multiplied  by  24,  giving  ||;  the  f  by 
12,  giving  ||  ;  and  the  |  by  8,  giving  |g. 

9.  §  +  §  =?  14.  |  +  |  +  |  -?  19.  2|+  |  +  |  =? 
10.  i  +  2=?  15.  i  +  |  +  i=?  20.5-i+6|+|=? 
11-T\+I=?  16-  7VH+.iV=?  21.  i+2|+T32=? 

12.  2  +  |=?    17.   |+V3+'l=?    22.     |+    §  +  -^=? 

13.  J  +TV=?     18.    |  +  I  +/3=?    23.    ^+  ^+  J  =? 
Answers:  2^,  1«,  1^,  Ii,  **,  i  12,  3|,  1|,  1      I/,, 


*This  is  simply  multiplying  both  terms  by  the  same  number. 
[Art.  69.] 


94  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


X.  THE  MERCANTILE  PROFESSION". 

THE  mercantile  community  may  be  divided  into  various 
classes  :  Importers,  Jobbers,  Wholesale  Dealers,  Commis- 
sion, Forwarding  and  Retail  Merchants,  Brokers,  etc. 

89.  Importers  purchase  goods  and   produce   in  foreign 
countries,  and  sell   them   in  the  home  market  to  jobbers 
and    wholesale    dealers.     They   also    receive    goods    from 
abroad  to  sell  on  commission. 

90.  Jobbers.     This    term   was  first  applied   to   persons 
dealing  in  stocks  to  a  limited  extent,  but  it  now  includes 
nearly  all  classes  of  wholesale  dealers.     We  speak  of  dry 
goods  jobbers,  produce  jobbers,  cattle  jobbers,  etc. 

91.  Wholesale  Merchants  or  dealers  buy  from   import- 
ers, jobbers,  manufacturers  and  producers,  and  sell  to  re- 
tail dealers. 

92.  Commission  Merchants*  act  as  agents  for  other  per- 
sons in  buying  and  selling  goods,  collecting  debts,  etc.,  for 
which  they  charge  a  percentage  on  the  whole  amount  of  sale, 
purchase  or  collection.     Merchants  of  this  class  usually  keep 
a  wholesale  department  in   their  warehouses,  where  they 
sell  their  own  goods  as  well  as  those  of  others,  and  even 
ship  merchandise  to  distant  places  for  sale  on  commission, 
thus  acting  in  the  capacity  of  principals  as  well  as  agents. 

The  person  who  sends  goods  to  another  to  be  sold  on  com- 
mission is  called  the  shipper  or  consignor ;  the  person  who 
receives  them,  agent,  correspondent,  consignee  or  factor ;  and 
the  goods  or  merchandise  sent,  shipment  or  consignment. 

93.  Forwarding     Commission    Merchants    and     Express 
Companies  are  intrusted  with  the  care  of  conveying  goods 

*  Auctioneers  belong  to  this  class. 


THE  MERCANTILE  PROFESSION,  95 

from  one  city  or  country  to  another,  for  which  they  charge 
a  percentage,  called  forwarding  commission.  This  class 
of  merchants  usually  have  warehouses,  and  on  rivers, 
wharf-boats,  for  the  storage  of  merchandise.  A  separate 
charge  is  made  for  all  goods  lodged  in  these  places,  ac- 
cording to  the  time  they  remain.  The  fee  charged  is 
called  storage. 

94.  Retail  Merchants  dispose  of  their   goods  in  small 
quantities  suited  to  the  wants  of  consumers. 

95.  Brokers  form  a  numerous  class.     They  assist  com- 
mission merchants  and  dealers  generally  in  finding  buyers 
for  their  wares,  and  trade  or  speculate   in   stocks,  lands, 
etc.     By  confining  their  attention  to  one  line  of  business, 
they  acquire  a  more  intimate  knowledge  of  its  details  and 
of  the  credit  of  persons  engaged  in  it,  and  are  thus  pre- 
pared to  render  valuable  services  to  both  buyer  and  seller, 
between  whom  they  act  as  middle-men. 

The  business  of  the  broker  does  not  require  the  invest- 
ment of  much  capital,  as,  unlike  commission  merchants, 
they  are  not  under  the  necessity  of  keeping  stores  or 
warehouses.  For  the  same  reason,  their  fees  are  smaller 
than  those  of  the  latter. 

There  are  Money,  Exchange  and  Bill,  Stock,  Custom- 
house, Heal  Estate  and  other  Brokers. 

MONEY,  EXCHANGE  AND  BILL  BROKERS. 

96.  They  buy,  sell  and    exchange  specie,  bills  of  ex- 
change, notes,  etc. 

The  entire  business  of  this  class  is  often  performed  by 
one  individual  or  company.  Being  considered  a  branch 
of  the  banking  business,  many  of  them  adopt  the  name 
of  bankers. 

Heal  Estate  and  Stock  Brokers  buy  and  sell  for  others, 


9G  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

lands,  houses,  stocks  in  public  funds  and  joint  stock  com- 
panies, etc. 

Custom-house  Brokers  find  employment  in  maritime 
cities,  by  assisting  masters  of  ships  in  obtaining  the  nec- 
essary papers  at  the  custom-house,  and  paying  duties  or 
taxes  incident  to  the  navigation  of  the  ocean. 

Ship  and  Insurance  Brokers  procure  freights  and  car- 
goes for  ships,  adjust  the  terms  of  charter  parties,  settle 
accounts  between  owners  and  masters  of  ships,  effect  in- 
surance on  ships  and  cargoes,  etc. 

Produce  Brokers  buy  and  sell  for  others  various  kinds 
of  farm  produce,  as  corn,  wheat,  cheese,  etc.  They  stand 
between  the  producer  and  dealer  or  shipper. 

The  business  of  other  brokers  will  be  sufficiently  indi- 
cated by  their  names. 

MERCANTILE  AND  COMMERCIAL  COLLEGES 

are  institutions  of  learning,  having  for  their  ostensible  ob- 
ject to  prepare  young  men  for  entering  the  mercantile 
profession.  They  are  got  up  by  private  enterprise,  some 
of  them  being  chartered  and  some  not.  The  chartered 
institutions  possess  no  advantages  over  the  others,  as  none 
of  them  have  the  power  of  conferring  degrees. 

The  course  of  study  in  this  class  of  colleges  usually 
comprises  instruction  in  book-keeping,  with  its  application 
to  the  various  branches  of  trade,  manufactures,  etc.;  mer- 
cantile arithmetic,  penmanship  and  business  correspond- 
ence ;  lectures  on  the  usages  of  trade,  negotiation  of 
business  paper  and  the  most  useful  branches  of  commer- 
cial law. 

When  conducted  with  ability  and  integrity,  commercial 
educational  establishments  rank  among  the  most  useful 
institutions  of  learning  of  the  day.  Though  of  compara- 


THE  MERCANTILE  PROFESSION.  97 

tively  recent  origin,  they  are  to  be  found  in  most  of  the 
larger  cities  of  the  Union,  and  receive  a  liberal  patronage 
by  all  classes  of  the  community.  Professional  men,  me- 
chanics, farmers,  and,  in  large  cities,  ladies,  are  to  be 
found  among  the  number  who  consider  their  education 
unfinished  until  they  have  passed  a  commercial  course  in 
one  of  these,  and  merited  a  diploma. 

It  may  be  safely  asserted  that  in  no  institution  of  learn- 
ing is  there  as  much  useful  information  obtained  in  so 
short  a  time,  and  at  such  trifling  expense,  as  in  commercial 
colleges. 

Business  men  may  obtain  a  knowledge  of  book-keeping, 
as  applied  to  their  own  business,  in  .a  few  weeks,  while 
most  youths  might  be  profitably  engaged  in  a  college  at 
least  one  year. 

PERSONS  EMPLOYED  IN  BUSINESS  HOUSES. 

The  persons  employed  in  mercantile  houses  are:  Book- 
keepers, Correspondents,  Solicitors,  Salesmen,  Travelers; 
Entry,  Bill,  Shipping  and  Engrossing  Clerks,  Junior 
Clerks  or  Boys,  Porters,  Coopers  and  Draymen. 

DUTIES  OF  THE  VARIOUS  OFFICES. 

Book-keeper. — The  book-keeper's  place  of  business  is  in 
the  counting-room.  His  duty  is  to  keep  the  accounts  of 
the  establishment,  in  a  variety  of  books  for  that  purpose, 
to  receive  and  pay  out  all  moneys,  and  deposit  money  in 
the  banks  for  safe  keeping,  to  make  out  bills,  render  ac- 
counts, statements,  etc.,  from  the  ledger  and  sometimes  to 
conduct  the  business  correspondence. 

Correspondent. — The  business  of  the  correspondent  is  to 
reply  to  letters  of  inquiry,  and  to  write  all  letters  of  bus- 
iness connected  with  the  house,  etc.  In  extensive  im- 
9 


98  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

porting  houses,  the  correspondent  is  usually  a  person  who 
understands  some  three  or  four  of  the  modern  languages. 

The  Second  Boole-keeper  assists  the  first  book-keeper. 
He  copies  or  transfers  to  a  journal  or  day-book  the  items 
found  in  the  sales-book,  journalizes  and  posts  to  the 
ledger.  It  devolves  upon  him  to  make  out  bills,  accounts, 
and  to  assist  in  the  counting-room  generally. 

The  Solicitor — traveling  agent — belongs  to  the  broker's 
class.  His  business  is  to  solicit  orders  and  secure  buyers 
for  houses  with  which  he  has  made  previous  arrangements. 
Accordingly  he  is  found  in  the  hotels,  on  the  steamboats 
or  at  the  railroad  depots.  When  he  finds  a  buyer,  he  con- 
ducts him  to  the  store  for  which  he  is  operating,  and,  if 
not  under  salary,  receives  a  commission  for  his  services. 

Clerk  is  a  general  term  applied  to  all  employes,  the 
porter,  drayman  and  cooper  excepted. 

Salesman. — The  duties  of  the  salesman  consist  in  un- 
packing, marking  and  arranging  goods  for  sale,  receiving 
customers  and  selling  to  them. 

In  some  houses,  the  salesman  receives  a  commission  for 
the  amount  of  trade  he  influences,  and  in  all  places  the 
amount  of  his  salary  very  much  depends  upon  this  cir- 
cumstance. 

These  statements  apply  principally  to  wholesale  busi- 
ness. 

The  Skipping  Clerk  receives  and  examines  goods  to  see 
if  they  agree  with  the  conditions  of  the  bill  of  lading,  and 
attends  to  the  shipping  of  goods  from  the  establishment. 
These  he  enters  in  a  book  for  that  purpose,  called  the 
shipping-book. 

The  Entry  Clerk  records  the  sales  made  by  the  sales- 
man in  a  book  called  a  blotter  or  sales-book. 

The  Bill  Clerk  makes  out  the  bills  or  outward  invoices 
from  the  sales- book. 


THE  MERCANTILE  PROFESSION.  99 

Entry  and  bill  clerks  should  be  rapid  penmen  and  ex- 
pert in  figures,  if  they  would  command  liberal  salaries. 

The  Engrossing  Cleric  assists  generally,  sometimes  in 
the  counting-room,  but  more  generally  as  entry  or  bill 
clerk.  He  is  simply  a  copyist.* 

Junior  Clerks  are  usually  boys  from  12  to  17  years  of 
age,  whose  duty  it  is  to  run  on  errands,  pack  and  unpack 
goods,  mark  boxes  and  packages,  and  keep  sale-rooms  in, 
order. 

After  acting  in  the  capacity  of  junior  clerk  for  two  or 
three  years,  they  are  promoted  to  more  lucrative  and  re- 
sponsible offices. 

The  youth  who  would  aspire  to  a  high  degree  of  use- 

fulness~in  his  profession,  should  not  rest  content  with  the 

•  I  / 1 

opportunities  for  improvement  afforded   by   the  store    or     *  j  J 

sales-room.  His  evenings  should  be  devoted  to  useful 
study,  until  he  acquires  proficiency  in  all  the  various 
branches  of  mercantile  business. 

The  study  of  Freedley's  excellent  "  Treatise  on  Busi- 
ness" is  highly  recommended,  especially  chap,  iii,  page  46. 

Heading-rooms,  libraries,  mercantile  associations,  mer- 
cantile and  commercial  colleges  are  to  be  found  in  most 
of  the  larger  cities  of  the  Union,  and  should  form  the 
places  of  resort  for  young  men  of  this  class  in  preference 
to  questionable  places  of  amusement,  too  much  frequented 
by  them. 

Porter. — The  business  of  the  porter  is  to  open  and  close 
the  store,  keep  the  store  and  counting-rooms  clean  and  in 
order,  pack  and  unpack  goods,  assist  in  handling  and 
weighing  heavy  goods,  marking  packages,  etc. 


~*In  many  houses,  the  whole  business  of  clerking  is  performed  by 
one  person,  while  in  others  many  more  offices  are  called  into  requi- 
sition than  are  noticed  here. 


r     '// 

m 


100          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

The  office  of  porter  is  a  more  responsible  one  than  most 
people  imagine.  By  the  faithful  discharge  of  its  duties, 
hundreds  of  men  in  this  country  have  been  placed  in  pos- 
session of  respectable  retail  houses. 

In  some  large  establishments,  there  are  two  or  more 
porters  engaged,  between  whom  the  duties  of  the  above 
are  divided. 

Cooper. — In  liquor  and  heavy  sugar  establishments,  pack- 
ing-houses, etc.,  the  services  of  a  cooper  are  required, 
whose  duty  it  is  to  open  and  close  hogsheads,  barrels,  etc., 
and  to  repair  damages  to  which  such  articles  are  subject 
in  carrying. 

Drayman. — The  drayman  acts  as  "carrier"  between  the 
etore  and  depot,  landing  or  wharf.  He  usually  keeps  a 
book  called  a  dray-book,  in  which  are  entered  the  con- 
tents of  each  load.  This  is  signed  by  the  clerk  at  the 
place  of  delivery,  and  when  the  entire  shipment  is  made 
the  amount  is  entered  in  the  Bill  of  Lading. 

XL  BILLS— INVOICES. 

97.  WHEN  goods  are  sold,  it  is  the  duty  of  the  mer- 
chant, or  one  of  his  clerks,  to  make  out   a    statement  of  ] 
the  quantity,  kind  and  price  of  each  article,  for  the  sat-  \ 
isfaction  of  the  purchaser,  and  to  enter  at  the  foot  of  such  > 
statement   the   whole   amount   of  the   purchase,  with    the ; 
payment  received,  if  any,  or  the  terms  of  settlement.     If  • 
the  goods  are  bought  to  sell  again,  this  statement  is  com- 
monly called  an  Invoice;  otherwise  it  is  called  a  Bill,  es- 
pecially by  the  purchaser. 

A  bill  or  invoice  is  sometimes  delivered  to  the  buyer  at; 
the  time  of  purchase;  but  it  is  usually  sent  with  the 
goods,  or,  if  the  buyer  resides  at  a  distance,  by  mail. 

An  invoice  should   specify  the  place  and  date  of  sale, 


THE  MERCANTILE  PROFESSION.  101 

the  name  of  buyer  and  seller,  a  description  of  the  goods, 
the  price  of  boxes,  etc.,  used  for  packing,  charges  for  in- 
surance, and,  when 'payment  is  not  made,  the  terms  of  sale. 
When  goods  are  received,  the  quality  and  quantity  are 
compared  with  the  invoice,  and  the  selling  prices  made 
out  from  it,  after  which  it  is  filed  away  or  pasted  in  a 
book  for  future  reference. 

98.  It  is   the   custom  of  merchants  to  have  their  bill 
heads  (heading  of  bills)  printed,  with  names  of  city  and 
street,  number  of  house,  and  such  other  matter  as  will  fa- 
cilitate the  labor  of  clerks,  or  otherwise  advance  the  inter- 
ests of  the  business.     A  specimen  form  of  such  heading 
will  be  given  in  the  bills  that  follow. 

99.  Filing  Bills.     When  there  are  many  bills  on  hand 
designed  for  collection,  they  should  be  folded  neatly  of  the 
same  length  and  breadth,  and  have  the  names,  addresses 
and  amounts  written  on  the  outside  at  the  top.     A  gum 
band  will  then  keep  them  firmly  in  their  places,  and  per- 
mit their  being  delivered  without  the  trouble  of  opening 
for  examination. 

100.  Retail    bills    are    rendered    periodically,    by    the 
month,  quarter,  half-year  or  year,  according  to  agreement 
or   the    usage   of  the    house.     When    a   pass-book  is   not 
kept,  it  is  well  to  have  a  memorandum  of  each  purchase, 
so  that  in  rendering  his  final  bill  the  merchant  need  not 
insert  the  items. 

101.  Account  or  Statement.     The  final  bill  of  a   mer- 
!  chant  now  goes  by  the  name  of  account  or  statement.     The 
!  head  contains  the  date  upon  which  it  is  drawn,  and   the 
!  word  "To"  substituted  for  "Bought  of."     In  the  margin, 
I  on  the  left,  are  the  dates  of  the  several  purchases,  with  tho 
|  words,  "For  amt.  rendered,"  or  "Amt.  pr.  bill  rendered." 

102.  In  making  out  bills,  the  three  requisites  are  ra- 
j  pidity,  legibility  and  accuracy.     The  principal  is  accuracy. 


102 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


In  business,  it  is  not  enough  to  be  right  after  one  or  two, 
or  perhaps  repeated,  attempts.  The  clerk  should  be  cor- 
rect the  first  attempt,  and  generally  is  so.  Boys  designed 
for  business  pursuits  ought  to  spend  much  time  at  bill- 
making,  until  they  acquire  familiarity  with  the  numerous 
abbreviations,  and  can  make  out  a  bill  from  dictation  al- 
almost  as  rapidly  as  the  items  can  be  called  off. 

103.  Finding  the  cost  of  a  number  of  articles  at  a  cer- 
tain price,  and  placing  the  amount  opposite,  is  called,  in 
bill-making,  extending;  adding  the  columns,  footing  up. 

104.  Receipt  on  a  Bill.     A  clerk  or  agent  may  write 
the  name  of  his  employer  to  a  receipt  and  it  will  be  good, 
if  he  write  his  own  initials  or  last  name  underneath. 

BOOKSELLERS  AND  STATIONERS'  BILLS. 

CINCINNATI,  June  16,  1866. 

MR.  HORATIO  NELSON: 

Bought  of  R.  W.  CARROLL  &  CO., 

PUBLISHERS,  BOOKSELLERS  &  STATIONERS,  WHOLESALE  &  RETAIL, 
117  WEST  FOURTH  STREET. 

TERMS :  In  making  orders,  be  particular  to  avoid  mistakes. 

All  claims  for  Errors  or  Damages  to  be  made  within  five  days  of  receipt  of  goods » 


1  Gro.  Pen-holders,  

2 

35 

*[  u  u  u 

75 

3  Doz.  Paper-Cutters,  .... 
2  "  Ebony  Rulers,  .... 
1  Rm.  Bill  Cap,  

2.00 
3.75 

6 

7 

7 

00 
50 
00 

2  «  Letter  Cap,  No.  1400,  .  . 
1  2-Qr.  Blank  Book,  .... 
3  Qr.  Blotting  Paper,  .... 

8.25 
40 
2.25 

16 

6 

50 

80 

75 

$47.55 

Rec'd  Paym't. 


R.  W.  CARROLL  &  Co. 
DAVIS. 


Gro.,  Gross;  Doz.,  Dozen;   Qr.,  Quire;  Rm.,  Ream. 


4 


BILLS—INVOICES.  103 

BOOKS— STATIONERY. 

105.  Books,  stationery,  etc.,  at  wholesale,  are  usually 
sold  by  the  dozen.     Paper  is  sold  by  the  ream,  bundle  or 
pound.     Writing  paper  is  put  up  in  half-reams;  printing 
paper  in  bundles  of  two  reams. 

In  the  exercises  which  follow,  the  teacher  will  find  it  to 
the  advantage  of  scholars  to  have  them  write  from  dicta- 
tion. The  bills  may  be  made  out  in  favor  of  the  learner 
or  otherwise. 

3.  J  dz.  Hooker's  Nat.  Philos.,  $14.40;  1  rm.  cap.  $6.50; 
1  do.  bill,  $7.00;  2  dz.!2-in.  ebony  rulers,  $3.75;  2  dz.  paper- 
cutters,  $2.00;  1  bx.  crayons,  35  c;  12  dz.  pass-books,  40  c 

4.  2  dz.  mucilage,  $2.75;    1   dz.  carmine,  $2.00;   2  dz. 
tin   cutters,  $2.00;    2  dz.  rulers,  $4.50;   -J-  rm.  natl.  cap, 
$5.50;  3  dz.  No.  4  pass-bks  44 c  £  rm.  treas'y  cap,  $8.40; 
1  qr.  blotting,  $2.25. 

5.  1  bx.  4560  5^  envelopes,  $2.25;  1    do.  8J  Manilla, 
$2.75;  1  dz.  Lincoln  portraits,  $3.00;  2  dz.  check-books, 
$12.24;    1000    penman's    blanks,   $20.50;    21T93    dz.    bill- 
books,  $2.16;   4   dz.   do.  2  qr.  ea.,  $2.50;    20  dz.  check, 
$2.16;   20  dz.  inv.,  $1.00;   20    dz.   day-books,  $1.80;   60 
dz.  journals,  $1.80;  20  dz.  ledgers,  $1.80;  sunds,,  $7.00. 
Credit — cash,  $210. 

6.  1  copy'g  press,  $10.00;  |  dz.  Rec.  of  a  Country  Par- 
son, $20;  TV  dz.  Wordsworth's  Poems,  $25.00;  T\  dz.  Be- 
ginning  Life,  $12.50;   y5^   dz.  Heads   and    Hands  in  the 
Wrld  of  Labor,   $12.00;   i£    Essay   on  Woman's   Work, 
12.50;  2T%  do.  Nat'n'l  Lyrics,  $4.80. 

Answers:  $55.80,  $30.21,  $33.75,  $31.02,  $150.16. 

SHOE  BUSINESS. 

106.  Shoes  are  usually  sold  at  wholesale  by  the  dozen, 
boxes  furnished  gratuitously. 


104          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Pairs. 

7.  12  Wos.   Goat  Tips  Bals 1.45 

6  Miss        "        "        " 1.75 

12  Childs     «        "        " 1.05 

12       «      Calf  " 1.05 

12  Wos.    Goat  Tips     " 2.50 

10      "      Last.  Lace  Cong 2.00 

6       «          "     But.  Gait 2.90 

6  Miss  Kid  Cong.       "     1.90_ 

Numbers 
on  boxes.  Pairs. 

8.  S     12  Child  Serge  Lace  Heel'd  Gaiters.      65 

474     13  Ladies  Kid  ch.  nl.  Balmorals 1.30 

533     12       "      Goat  D.  S.        "  1.28 

2386  11  Wos.  Kid  Cong.  D.  S.  Boots....   1.374 

2449  12       "       Serge  Cong.  Gaiters 1.20 

2593  12       "  "         "  "      1.124 

2475  24  Child  Gai.  Peg-Heel'd  J.  L.  Boots      45 

2586  12       "     End.       u  "  cop.  tip.      45 

2580  24      "     Buff       "         Lace     "     "         40 

2575  12  Miss  GaL         "         J.  L 674 

2413  15     "      Goat       "         Cong  Gait...      95 

2517  12     "         "          "  "  . . .      60 

2591  9  Ladies  Goat,  tip.  ch.  nl.  Bals 2.00 

2431  12  Youths  Buff  Peg-H'1'd     "     724 

2461  12  Ladies  Calf          "         tip  Bals..   1.25 

2588  12       «       Kid  ch.  H'l'd  «   . .    1.50 

2589  12       «       Peb.  Calf  ch.  HTd  tip  Bals  1.50 

2590  12  Miss  Kid  ScT.  Welt  "     1.05 


ABBREVIATIONS. —  TFbs.,  Womans;  D.  5.,  Double  soled  ;  Cong.,  Con- 
gress; Grd.j  Grained;  End.,  Enameled;  J.  L.,  Jenny  Lind;  Cop* 
Tip.,  Copper  tipped;  Ch.  NL,  Channel  nailed;  Hid.,  Heeled;  Sd, 
Wit.,  Sewed  Welt;  Peb.  Cf.,  Pebbled  calf;  Bals.,  Balmorals;  But. 
Oait.,  Button  Gaiters. 


BILLS— INVOICES.  105 

MILLINERY. 
9.  60  12  15-Braid  Bonnets @  J0.62J 


68     6 

it 

1 

.25 

70    4 

7-Braid 

a 

u 

1 

.50 

.  80     2 

7     " 

a 

u 

3 

.00 

86     2 

7     " 

n 

u 

3 

.75 

6 

PCS. 

No.    1  Tafft.  Ribbon  

a 

15 

5 

a 

"      2 

u           u 

a 

28 

3 

a 

"      4 

u           u 

u 

48 

2 

u 

"      6 

it           it 

u 

75 

1 

u 

"    12 

u           u 

tt 

1 

.10 

3 

tt 

Bonnet  Ribbon  

u 

2 

.00 

2 

a 

u 

u 

a 

2 

.50 

3     1 

Box 

Ruches.  . 

a 

1 

.50 

415     1 

ti 

« 

u 

2 

.25 

210    i 

Doz. 

Bunches 

Flowers  

u 

18 

.00 

J     "  «          Feathers "  36.00 

1  PC.  Black  Silk,  20  yds "        87J 

GROCERY  BUSINESS. 

107.  TARES. — In  Cincinnati:  New  Orleans  sugar,  in 
hogsheads,  10%;  Cuba  and  Porto  Rico,  12%;  sugar  in 
boxes,  15%.  Rice  in  tierces,  10%;  indigo  in  ceroons, 
11%;  in  boxes,  actual  tare;  salt  in  barrels,  each  30  Ibs.; 
coflee,  cotton,  spices,  feathers  and  salt,  in  bags  and  bales, 
no  tare;  manufactured  tobacco  in  kegs  and  boxes,  (not 
enumerated,)  actual  tare;  madder  in  casks,  actual  tare; 
lard  and  bacon  in  packages,  actual  tare.  Lard  kegs  tared 
after  being  emptied.* 

*  Butter  in  firkins  is  subject  to  an  allowance  of  2  Ibs.  for  soakage; 
in  rolls  packed  in  half  barrels,  1  Ib.j  barrels,  2  Ibs.;  tubs— 50  to 
70  Ibs.— 1  Ib. 


106          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

In  New  York:  The  tares  differ  slightly  from  the  above; 
"but  the  list  being  a  long  one,  it  is  reserved  for  the  revised 
edition  of  "  Nelson's  Mercantile  Arithmetic." 

Extra  Charges  may  be  made  for  drayage,  insurance, 
cooperage,  storage,  boxes,  bags,  etc.,  in  which  goods  are 
packed  in  the  store.  No  extra  charge  is  made  for  the 
original  package.  In  the  bills  which  follow,  some  of  these 
charges  are  introduced. 

10.  1  Hhd.  N.  0.  Sugar,  *^oo  **1080  Ibs.  .@$.07 
4  Brls.  N.  0.  Molasses/H  169  gals  ----  "     35 

1  Trs.  Eice,  ™g  630  Ibs  .............  "       4 

20  Bags  Rio  Coffee,  3200  Ibs  ..........  "     11 

2  Half  Chests  Black  Tea,**  o1^**  72  Ibs.  "     25 

1  00—28 

3  "  "     Yng.  Hyson  do.  150  Ibs.  .  "     50 

1  "  "     Imperial        do.    60  Ibs.."     40 

2  "  "     Gunpowder  do.  110  Ibs..  "     60 
1     "  "     Oolong  Blk.  Tea,  45  Ibs.  "     40 


VJ 

1 

Box 

tiAWAUVt.        V^  A  U.J-iC*il.iJ.  \_fJLi 

5  lump  Tobacco, 

ll\  108  Ibs.. 

cc 

25 

1 

a 

pound  lump  " 

'4J  124  Ibs.. 

(C 

20 

1 

u 

Va.  pound    " 

»4o  120  Ibs.. 

a 

35 

1 

a 

8  lump           " 

J|g  125  Ibs.. 

u 

22 

20 

Brls 

Rect.  Whisky 

800  gals  

a 

17 

4 

a 

Ginger  Wine, 

160  gals  

a 

60 

\ 

Cask 

French  Brandy, 

40  gals  

a 

4 

.00 

% 

a 

Port  Wine, 

45  gals  .... 

a 

2 

•00 

10 

Brls. 

Bourbon  Whisky, 

405  gals  .... 

a 

1 

.00 

J 

Brl. 

Holland  Gin 

20  gals.... 

u 

1 

.50 

5 

%  °* 

F  , 

*  Gross  Weight.  J  Gallons  in  each  barrel. 

•{Tare,  or  weight  of  bag,  box,  etc.         **Net  Weight. 


BILLS-INVOICES.  107 

The  small  figures  on  the  left  indicate  the  prices  of  boxes, 
barrels,  etc. 

11.  100  Boxes  Cheese,  4'?<>  3690 @$  .08 

30  Firkins  Butter,  3f<j<>  2820 "       15 

100  Boxes  $2  °  Starch,  4810 "         5 

100       "      ***  Star  Candles,  4000 "       20 

20  Bbls.  $*5  Lard  Oil,  810  gals "       85 

50     "      Mess    Pork "16.00 

10  Tierces  S.  C.  C.  Hams,  3|f  g  3000.   "       11 

30  Kegs  Lard,  'J'g  1334 "       12  J 

15  Bbls.  Mess  Beef «  15.00 

Com.  for  purchasing,  $1521.75 "  Z\% 

Drayages 16.00 

Insurance  on  $5000 59.88 

12.  1  bag  Pepper,  103 10J 

1     "    Allspice,   128.. 10£ 

4  dz.  Shakers'    Brooms 2.40 

1  Em.  Cap  Paper 4.00  W%  off. 

1     "     Med     "     6.00        "  " 

1     "     D.  0     "      8.00        "  " 

5  bxs.  10  Ger.  Ex.  Soap,  297.        7 

1  Keg  Soda,  112  Ibs 5J 

1  bx.  20  Saleratus,  61  Ibs....        5J 
1     "  20  Saltpeter,  47    u   ....        9J 

Drayage 79 

13.  10  Bbls,  «•«-.<>  Sugar, 

246  23  245     20 

233  18  246     17 

250  21  275     21 

227  22  232     19 

239  21  266  _25 

2459  207=2252  lbs.@12|c,  $ 
Drayage 1.60 


108          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

14.  1  Bbl.  ' 35  Lard  Oil,  41 85 

1  Hf.  bbl.  75  98%  Alcohol,  21£ 54 

1  Dz.  Washboards 2.25 

1  '  °  Sk.  Bar  Lead,  96  Ibs 6J 

1  Pkg.   Yarn,  28  Ibs 26 

2  Doz.  No.  2  Brooms . . . , 2.40 

1  Trc.  Ex.  S.  C.  Drd.  Beef  2™ 25 

1  Ke<r  lOd  Nails 3.00 

|  Bbl.  5°  Lard  Oil,  23 95 

Drayage 7«5 

15.  2  Cases  H.  Cheese,  46  Ibs 28 

4  "      S.  W.  R.  Cheese,  !?|  112  Ibs..  22 

3  "      E.  D.            "          87    78  u  -fl  18 

2  "      N.  W.  R.     "          25    21  "  ..  17 

1  Bbl.  Cuba  sugar,  2|f  202  Ibs 13  J 

4  Boxes  MR.  Raisins 4.40 

2  £      "        "         "        2.40 

2  i      «        "         "        .. 1.45 

3  Jars  Prunes 45 

5  Drums  Figs,  37^  Ibs 24 

1  Case    100  half  boxes    Sardines 42 

1  Box  American  Castile  Soap,  28  Ibs..  16* 

1     "     French             "         "       35    "   ..  16 

1  Doz.  Cox  Sparkling  Gelatine 6.00 

2  Boxes  *°  Saleratus,  128  Ibs 7^ 

5  Half  boxes  Star  Candles,  100  Ibs 24 

1  Bbl.  25  Com.  Smok.  Tobacco,  100  Ibs  12 

1  "     2S  S.  F.     "            "            83    "  25 
•  2  Boxes  40  Starch,  96  Ibs 7 

2  "            Soda,      92    «   13 

Sk.,  sack ;  Pkg.,  package ;   W.  R.,  Western  Reserve.    2  £  boxes 
raisins,  2  half  boxes. 


BILLS— INVOICES.  109 

16.  1  Bbl.  25  Soft  Refined  Sugar,  240  Ibs. .  @     16 
1     «     25  Hard     »  "      220    «    ..   "      17 
1     "     25  Granulated          "       212   "    ..   "      25 

1  Sack  Java  Coffee,  136  Ibs "      41 

1      "     Rio         "       156    " "     28 

1      "     Laguayra,  111  Ibs "      30 

|  Chest  Y.  H.  Tea,  82—14^68  Ibs...  "  1.25 
I     "       Black     «      53— 12=41  Ibs...  "  1.35 
J     "       Y.  H.  Canton  Tea,  f  *  68  Ibs. . .  "      90 

i     "  "      Fine,  ?o  66  Ibs "1.60 

-J     "       Oolong  Common  Tea,  }$  41  Ibs.  "      90 
1  Sack  African  Peanuts,  2  bushels ....       2.40 

1     "      Roasted  do.,  3  bushels 3.10 

1  Bbl.  25  Dried  Peaches,  « }  J  95  Ibs.  . .  «      21 
1     "     25      "      Apples,  105  85  Ibs....  "      15 

1  Box  20  German  Soap,  64  Ibs . ..  "      12£ 

1     "      2  °  Common     "      64  Ibs "      12 

1     "      20  Cincinnati  Extra  do.,  61  Ibs. .  "      14 

1  Bbl.  *5  Prm.  Wh.  Beans  ~?j>  216  Ibs.  "        2-f 

2  Boxes  2  doz.  Baking  Powder "  4.80 

17.  2  Sks.  Rio  Coffee,  323 25 

1  Bbl.  Molasses,      45—  J 60 

1  "     37  Ricej       235—19 10 

2  "     75  Sugar,     iJl=?g 12 

Drayage 50 

DISCOUNT. — An  abatement  entitled  discount  is  often  made  on  the 
bill  for  cash  or  when  goods  have  fallen  in  price.  When  making 
such  abatements,  the  clerk  should  remember  that  he  is  discounting 
the  profits  as  well  as  the  first  cost.  For  instance,  I  buy  goods  at 
$100,  and  sell  them  at  a  profit  of  50  per  cent.,  which  makes  the 
price  $150.  Now,  if  this  is  discounted  at  40  per  cent.,  it  does  not 
follow  that  a  gain  of  10  per  cent,  is  made.  40  per  cent,  of  $150= 
$60,  which,  taken  from  the  selling  price,  leaves  $90,  making,  in- 
stead of  a  gain  of  10  per  cent.,  a  loss  to  that  amount. 


110          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

18.  20  Bbls.  Molasses, 

43J  44  43 

42J 


44J 

41—1 

44^ 

43* 

44 

441 

44 

42^ 

42J 

43 

43J 

41—1 

44 

43 

865J—  *2@75o 

2.00 

19.  10  Tcs.  Shaffer's  Hams  3*'<> $0.21 

5     "     A.  Spence's  S.  C.  do.  ^Jf 25 

1  Bbl.  25  20  dz.  Tongues 1.00 

Com.  on  $1013.78 2%% 

Exch \% 

Drayage 2.00 

Answers:  $86.24,  $262.26,  $1046,23,  $138.73,  $285.60, 
$185.51,  $4152.87,  $1665.97,  $649.63,  $688./5,  $131.90, 
$228.74,  $130,  $97.09. 

NOTE. — When  merchants  can  not  fill  the  orders  from  their  store, 
they  sometimes  buy  elsewhere,  and  charge  their  customer  cost  and 
a  commission,  as  in  the  last  bill. 

The  Teacher  should  not  confine  himself  to  the  few  exercises  in 
bill-making  given  in  this  work,  as  he  will  find  it  both  interesting 
and  profitable  to  the  learner  to  represent  other  kinds  of  business  in 
the  same  way.  Any  number  of  bills  can  be  dictated  from  price 
lists  or  prices  current,  to  be  had  in  any  large  city. 
*  Leakage. 


"^   , 


BILLS— INVOICES.  HI 


DRY  GOODS  * 

Yds.     Price 

20.  1  PC.  H.  A.  g-  Blea.  Muslin 45  25 

1    "    A  |  Light  "      43  15 

1  "    D.  Q.  |  Fine  Bro.  Muslin. . . .  43  22 

2  "    P.  I  Muslin 702  24 

1    "    L.  Check 681  26 

1    "    W.Jeans 462  30 

A3"    Fey  Prints 1261  15 

B    1    "      "          "      372  16 

C    9    "      "          "      3552  17 

1    "    BuffChambray 202  35 

1    "    Dom  Gingham... 35  25 

1  "    L  "        602  25 

2  Doz.  Coats'  Spools 1.10 

1     "     |  Hose 3.00 

1     "     Ladies'  L.  W.  Hose 6.75 

No.4    1  Pk.  Pins 70 

31"       "..... 75 

21"       "     90 

1  Doz.  Lin.  Hdkfs 6.00 

1     "      H.  S.     "      9.50 

f  1  PC.  Sheetgs  to  fill 352  24 

2  Bxs  5  °  and  Strapping  5  ° 2.00 

*  Though  practically  correct,  some  of  the  answers  to  these  bills 
will  be  found  mathematically  wrong.  Accuracy  in  cents  has  been 
sacrificed  in  conformity  with  business  usage,  which  often  rejects 
fractions  in  extensions,  alternately  adding  a  cent  and  rejecting  a 
half  or  fourth.  The  letters  "H.  A."  etc.,  indicate  the^rade  of  nius- 
liiis,  or  are  the  initials  of  the  maker  or  factory ;  the  figures  and 
letters  in  the  margin  are  marked  in  the  wholesale  house  to  distin- 
guish lots  of  nearly  the  same  kind  from  one  another. 

tNot  in  the  order;  but  will  not  be  objected  to  by  the  buyer,  as  it 
prevents  goods  from  shifting  in  the  case. 


112          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


21.  4 

PCS, 

,  Prints  

.    1853 

17 

1 

tt 

Jeans  

.      451 

30 

1 

a 

Cottonade  

.      381 

37s 

5 

it 

Chamb.  Gingham  

.    1542 

35 

2 

it 

Luster  

.    1012 

30 

2 

Doz 

.  Skirt  Braid.  

1 

.00 

10 

u 

Coats'  Spools  

1 

.10 

No.  101 

u 

Miss's  Hoops  

5.00 

181 

M 

a         tt 

9.00 

201 

u 

n         it 

10.00 

30  4 

a 

2  Gore  Miss's  Hoops  

2 

16 

.00 

22.  1 

PC. 

Fancy  Cassimere  

27f 

2 

.20 

1 

u 

French         "         

28| 

2 

.85 

1 

a 

Mixed          "        

31 

2 

.90 

1 

M 

Amoskeag  A  Ticks  

GO 

47 

1 

U 

Glassgow  Gingham  

40J 

27 

1 

u 

Victoria           "         

37 

42 

1 

a 

Shawmut  Can.  Flannel  .... 

35J 

342 

3 

u 

LX.L.Wl.do.41,64,34.. 

139 

73 

1 

u 

Roan.  Striped  Shirting  .... 

37J 

31 

2 

a 

Fey  Slk  Dress  Goods,  jj.. 

71 

1 

.10 

1 

tt 

Village  Green  Checks  

37| 

23 

4 

tt 

Lonsdale  Checks   f?   ?<)••• 

143 

21 

1 

it 

Blk  French  Broadcloth  .  .  . 

29} 

4. 

.25 

1 

tt 

English                   " 

37J 

5 

.00 

2 

tt 

Bolivar  Denims  \  g  

98 

20* 

1 

« 

Charter  Oak  Denims  

60 

22 

1 

a 

38| 

17a 

2 

IK 

Duchess  Delaines,  311  361  .  . 

67* 

30 

4 

a 

Semper  Idem,  40  38  271  36. 

1151- 

322 

2 

a 

Barre  Brown  Sheeting  \  g  .  . 

76 

34 

1 

tt 

Great  Falls  do.  do..  

42J 

28 

2.50 

BILLS— IN  VOICES.  113 

23.  75     48  Doz.  Gent's  Shakspeare  Coll 2.00 


6     60 

u 

Ladies  Plea                "     , 

1.26 

1     48 

N 

"        BvronEmb.  "    , 

1.50 

1     45 

U 

"        Gar.        "       "     , 

1.50 

1     48 

it 

u          per            a         tt 

1.50 

1  120 

tt 

"        Sq.  Gar.  C.&C.E. 

Coll. 

1.00 

1     29 

a 

"        Mull.  Edge  Gar, 

u 

1.50 

1     20 

tt 

Vic.  Cuffs,  2  C  ., 

2.50 

1       2 

a 

"      ..Octagon  C.&C.E 

.Sets 

4.00 

1       5 

N 

"         Sqr.  Emb. 

u 

5.50 

1       8 

(C 

"        Point  Wht.  Trim.  " 

4.50 

16    47 

n 

Shirt  Fronts  

3.50 

24.  11  10  PCS. 

4 
4 

Shirting  Linen  

294 

$0.37 

12  10 

a 

4 
4 

tt                       it 

297 

40* 

13  10 

tt 

4 

it                « 

300 

44 

14  15 

(1 

4 

u                 « 

433 

49 

15  15 

u 

A 

u                u 

447 

55 

16  10 

tt 

1 

U                       It 

285 

60 

17  10 

« 

4 

tt                       U 

287 

85 

25.  1342    2 

Doz. 

Napkins  

2.75 

1343    2 

u 

u 

3.40 

1356    2 

a 

4.50 

1270    2 

u 

it            tl 

4.00 

1138    4 

Felt 

Table  Covers  

2.85 

1416    1 

PC 

fl62 

57* 

1417    1 

tt 

"           "          "3  

26 

622 

1418    1 

a 

"          u          "    4.  . 

26 

70 

1419    1 

n 

tt              u             a      /) 

27 

75 

1421    1 

u 

u              «             a      ^ 

26 

872 

1422    1 

tt 

a              u             a     g 

27 

1.00 

Emb.,  Embroidered;    Gar.,  Garote;    Per.,  Persiguey;    C.  $  2    <?., 
Cord  and  2  Cord;    Vic.,  Victoria. 
10 


114          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
QUEENSWARE. 

MV« 

3  Doz.  Edged  Plates $0.40 

10     "         "           "      50 

5     "     *CC         "      50 

£     "         "       Dishes,  ea,  $1.75,  2.25,  2.75. 
1     "         "       Bakers,  ea.  $1.50,2.00, 2.50. 

1  "         "       Beaded  Nappies,  ea.  1.75, 2.25 

|     "         "       Tureens 3.50 

2  "         "       Bowls 80 

2J  "         "           " 60 

3  "         "           "      50 

\     "         "       Pitchers 3.50 

1     "    Colored       "       ea.  $2.50,  $4.00 

4  "         "       Bowls 872 

5  "         "           " 65 

6  "         "           " 55 

9  Sets  CC  Teas 20 

36     "     Painted  Teas 20 

Crate 1.00 

27  HAT  BUSINESS. 

\  Doz.  Men's  Black  Cass.  Hungarian 21.00 

\     «         "           "         "             "          27.00 

6J     "         "           "         "      33.00 

|     "         "           "       Broad  Brim  Wool. ..  14.00 

1     "         «           "       Wove  Senate 12.00 

J     "         "            "       Cashmerette 15.00 

\     "     Boys'         "        Hungarian 7.00 

4     "         "        Caps,  assorted 12.00 

i     "         "           "         "          9.75 

J     "         "        Cloth  Caps 9.00 

\     "     Children's  Fancy  Caps 8.00 


*  CC,  cream  colored. 


i 


BILLS— INVOICES.  115 


28  HARDWARE. 

25  Bars  1  JX  J  Bar  Iron  .........   750  Ibs.  $0.03 


6  Bund.  §  Round     "     .........    625    "  4 

4     «       11  x|  Dandy  Tire  Iron..    500    "  3s 

3  "       1    X|  Horseshoe  Iron^ 

4  "       1   XT7e         "            "      [  •  975    "  4 
2     "       1   X  -\         "            "3 

2  Sets  1JX4  L.  Pitts  Springs^ 

3  »     1JX5  "       "           "        t-  500    «  10» 

2  "     2  v5  u       "  u 

1  "     2   X8  ".      "  " 

4  Slabs  14  X  |  Steel  ...........  .600    "  72 

3  Doz.  Amos  No.  2  Shovels  ............  10.50 

2  "     Rowland's  No.  2  Shovels  .........  8.50 

1     «             "          No.  2  Spades  .........  8.50 

6     "      No.  5684  Pocket  Knives  .........  7.50 

3  "       "     4215       "             "       .......  4.50 


Answers:  $238.88,  $160.39,  $56.73,  $194.92,  $1234.04 
$832.60,  $232.53,  $981.51,  $318.25,  $735.63. 

MERCHANT  TAILOR'S  BILL. 

9-  NEW  YORK,  Ap.  5,  1867. 

MR.  A., 

To  B. 

Jan.  3.  For  1  Blue  Beav.  Overcoat 70.00 

Feb.  9      «     1  Pr.  Blk  Doeskn  Pants 23.50 

$93.50 
Mar.  2.  Cr.— By  Cash 50.00 

$43.50 

To  THE  TEACHER. — The  learner  ought  to  be  taught  that  bills  are 
seldom  punctuated,  and  that  even  the  dollar  or  cent  signs  or  @  are 
rarely  used.  It  is  proper,  however,  to  insert  the  dollar  sign  at  foot- 
ings. 


116          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

FARMER'S  BILL  WITH  CREDITS. 

RICHMOND,  Nov.  1,  1869. 
MESS.  TRADER  &  Co., 

To  JONATHAN  FARMER. 

Jan.  3.  For  3  Durham  Heifers 25.00     75.00 

"      "    1  2-yr.  old  Bay  Colt 50.00 

Ap.  11.     "    1402  Bu.  Corn 50     70.25 

"      "    52  Dz.  Chickens *  1.372      7.561 

9.     "    1232  Ibs.  Butter 183    23.15* 

Cr.  "$225^97 

Ap.  4.  By  25  Hogs,  3147  Ibs  32 110.142 

9.    «    1  Plow 7.622 

"    Cash 108.20  $225.97 

JONATHAN  FARMER. 

MECHANIC'S  BILL. 

ST.  Louis,  July  3,  1867. 
MESS.  BUREAU  &  Co., 

To  JOHN  WORKMAN. 

Jan.  1.  For  Cab't  W'k  on  3  dz.  hf-Fr.  Bed's.  20.00     60.00 
Feb.  9.     "         "       "      "   4"    2-pn'ldo...   30.00  120.00 

Cr.  180.00 

Jan.  1.  By  Cash  on  Ace 10.00 

"      8.     "   .Order  on  W.  Shoemaker f 15.00 

"    15.     "    Cash 20.00      45.00 

Settled  by  note  at  90  ds $135.00 

JOHN  WORKMAN. 

When  the  signature  of  the  merchant  is  required  to  a  part  pay- 
ment, the  phrase  Balance  due  should  be  written  opposite  the  amount 
unpaid. 

*  Express  the  1372  in  mills  before  multiplying. 
t  For  form  of  order  and  note  see  index. 
Hf.  Fr.  Bd's.>  half  French  bedsteads;  pnL,  panel. 


\ 


REDUCTION.  117 

DRESS-MAKER'S  BILL. 

BOSTON,  Sep.  9,  1866. 
MRS.  AFFLUENT, 

To  M.  E.  FASHION. 

For  Making  1  Moire  Antique  Dress 25.00 

"     Trimming  do 75.00 

$100.00 


XII.  COMPOUND  NUMBERS. 

BESIDES  the  distinction  made  between  numbers  in  chap- 
ter viii,  they  may  be  divided  into  simple,  and  compound. 

108.  A  simple  number  may  be  abstract  or  concrete,  of 
one  denomination,  as  27  men,  35  dollars. 

109.  A  compound  number  is  always  concrete  and  com- 
posed of  more  than  one  denomination,  as  157  dollars  50 
cents,  29  pounds  14  shillings  and  6  pence — two  numbers, 
each  expressing  one  sum  of  money. 

110.  Reduction  of  compound  numbers. 

111.  REDUCTION  is  the   process  of  changing   concrete 
numbers  of  one  denomination  to  those  of  equal  value  in 
another.     If  I  multiply  5  bushels  by  the  number  of  pecks 
in  a  bushel,  I  reduce  them  to  pecks,  and  thus  change  both 
the  number  and  denomination,  while  I  preserve  the  value.* 

112.  Changing    the    denomination   from  a  higher   to  a 
lower,  as  bushels  to  pecks,  is  called  reduction  descending; 
while  the  reverse  process,  as  changing  pecks  to    bushels, 
is  called  reduction  ascending. 

A  few  examples  will  suffice  to  teach  all  that  is  neces- 
sary to  be  known  on  this  subject. 

*  On  page  70  are  several  exercises  which  properly  belong  to  this 
subject. 


118          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

1.  To  reduce  35  acres  to,  or  represent  them,  in  square  feet. 

35  Acres  EXPLANATION. — In  1  acre  there  are  4  roods, 

4  in  35  acres  there  are  35  times  4  roods  (35X4) 

TT7  r>  7  or  140  roods;  in  1  rood  there  are  40  square 
rods,  in  140  roods  there  are  140X40  or  5600 
sq.  rods ;  in  1  sq.  rod  there  are  30J  sq.  yards, 


5600  Sq.  Rods    jn  5^00  Sq.  rods  there  are  5600X30J  or  169400 
sq.  yards ;  in  1  sq.  yard  there  are  9  sq.  feet, 


30j 


168000  in  169400  sq.  /ards  there  are  169400X9  or 

1400  1524600  sq.  feet. 

REMARK. — It    will    be  observed    that    the 
game  regulfc  would  haye  been  produced  by 

multiplying  the  35  by  43560,  the  number  of 


1524600  feet  in  an  acre. 

When  there  are  items  between  the  highest  and  the  low- 
est denominations,  they  should  be  added,  as  shown  in  the 
following  example: 

2.  In  £75,  13s,  6Jd  how  many  farthings? 

75  13  6J 
20 

1500  Shillings  in  £75. 
13  Shillings  added. 

1513   Whole  number  of  shillings. 
12 


18156  Pence  in  1513  shillings. 
6  Pence  added. 

18162  Sum  of  pence. 

4  Par  things  in  1  penny. 

72648 

9. 


72650  Parthings  in  18162J  pence. 

To  add  the  items  mentally,  the  3  shillings  would  occupy  the  place 
of  the  first  0;  then,  multiplying  the  5  by  the  2,  we  would  obtain  10 
to  which  might  be  added  the  1,  making  11;  then,  multiplying  the 
7  by  2,  we  get  14  and  the  one  carried,  making  15  or  1513  at  once. 


REDUCTION.  H9 

3.  In  3  miles  21  rods  how  many  yards? 

4.  In  145  tons  25  Ibs.  of  hemp  how  many  pounds? 

5.  How  many  pence  in  £197,  17s,  9d? 

6.  How  many  farthings  in  £57,  13s,  6£? 

7.  In  93  barrels  of  apples  how  many  pecks,  each  barrel 
containing  2  bushels  3  pecks? 

Answers:  324825,  55370,  47493,  1023,  5395£. 

113.  To  reduce  concrete  numbers  of  a  lower  denomina- 
tion to  those  of  a  higher,  the  process  will  be  the  reverse 
of  the  last. 

8.  Reduce  72650  farthings  to  pounds. 

4)72650  Farthings.         In  farthings  there  will  be  one  fourth 
1  9M  m  t\9      9    P  as  many  Pence;   in  pence  one  twelfth  as 

~1' —  many   shillings;    and    in   shillings  one 

2 1  0)151 1 3 — 6  Skill.       twentieth    as    many    pounds.     The    re- 
£75      13      61  raainders  are  13  shillings,  6  pence  and  2 

farthings  or  J  penny. 

9.  Reduce  4163  linear  inches  to  yards. 

10.  Express   31456739   minutes   in   years,   months    and 
days,  allowing  365  days  6  hours  to  the  year.* 

11.  Reduce  456372  farthings  to  pounds. 

12.  At  1  mile  in  4J  minutes,  how  many  miles  would  a 
locomotive  run  in  5  hours? 

13.  A  ship  sailed  3000  miles  in  16  days;  what  was  her 
average  speed  per  hour? 

14.  In  35  cubic  yards  how  many  cubic  inches  ? 

15.  In  5J  square  rods  how  many  square  feet? 

16.  Divide  a  log  55  feet  in  length  into  15  equal  -parts, 
and  express  the  result  in  feet  and  inches. 


*  Reduce  the  year  to  minutes  and  divide  them  into  the  minutes 
in  question,  which  will  give  the  number  of  years.  The  remainder 
being  minutes,  may  be  reduced  to  hours,  etc.,  as  in  reduction. 


120          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

17.  What  will  be  the  cost  of  35f  bushels  of  strawber- 
ries at  15  cents  a  quart? 

18.  At  5  cents  for  3  sheets  of  paper,  how  much  money 
can  be  obtained  for  35|  reams,  allowing  half  price  for  the 
outside  half  quires  (35-J-  qrs.)  of  each  ream? 

19.  A  merchant  buys  15  barrels  of  potatoes,  containing 
2  bushels  2f  pecks  each,  at  $1.25   per  bushel,  and  sells 
them  at  25  cents  a  half  peck;  how  much  does  he  make, 
allowing  \  a  peck  per  barrel  for  loss  in  measuring? 

20.  In  316754  Ibs.  of  hemp,  how  many  tons,  cwt.,  etc.? 
Answers:  141,  8,  18;  17160;  3,  8;  66| ;  945;  27690;  59, 

9,  25,  4,  59;  49|;  475,  7,  9;  7jf ;  26.48T\;  1497|;  1632960. 
114.    To  add  compound  numbers. 

1.  What  is  the  amount  of  the  following  sums  of  British 
money  ? 

£       s.       d.         SOLUTION. — 1.  We  first  add  the  fractions,  calling 

18  17      4^r     them  farthings,  which  makes  6  farthings;  these  we 

19  g      7JL     reduce  to  pence  by  dividing  them  by  4.     f— If  op 

17        7      83-     *-'     Wr*te  ?  ancl  a(1(*  tlie  *  Pennv  to  tne  column  of 

5     pence,  which  makes  20  pence;  this  number  divided 

55  11  8-J-  by  12  (the  number  of  pence  in  a  shilling)^!  shilling 
and  8  pence.  Write  the  8  under  the  pence,  and  add  1  to  the  units 
of  the  shillings'  place,  which  makes  21;  write  1  and  add  the  2  to 
the  ten's  column=3  or  31  shillings,  which,  divided  by  20— £1  and 
11  shillings  left.  Write  the  latter  under  the  shillings  and  add  tho 
1  pound  to  the  pound's  column=r£55.  Ans.  <£55,  11s,  8J. 

Add  the  following: 

2.  £17  18  llf  +  £14  17     2J+  £16  14  8  =? 

3.  £17  19     OJ+  £45     0  11|+£111  10  2J=:? 
t.  £116  16     6  +£320  14     5{-+  £38  18  8  =? 

5.  £24  18  6  +  £180  10     Of+  £66  19  11|=? 

6.  £175  19  7|+     £90     8     8|+£575  12     6£=? 

7.  £201  17  6|+£1010  10  10]+£970  19  11|=? 
Totals,  £3297  17  9f  £700,  10s,  8d. 

Hhtftk^- 


COMPOUND  NUMBERS.  121 

115.   To  subtract  compound  numbers. 
1.  From  £19,  4s,  4d  take  £14,  7s,  6]d. 

£  s.  d.  EXPLANATION.  —  We  can  not  take  J  from  nothing, 
19  4  3  so  we  add  a  penny  to  both  terms;  subtracting  -J-  from 
14  7  6^  the  1  penny,  or  4  fourths,  we  have  Jleft.  Adding  Id 
^T"T^  ^  to  the  6d  we  have  7d,  which  we  can  not  subtract 
•*  from  the  3d  above,  and  accordingly  add  Is  to  both 
numbers.  7  from  Is  3d  or  lod,  leaves  8d.  Adding  Is  to  the  shil- 
lings, we  have  8s,  which  can  not  be  taken  from  4s  without  adding 
£1  to  both  numbers;  £1  to  4s=24s;  8s  from  24s=16s.  Then 
adding  £1  to  the  14,  we  have  £15,  which,  taken  from  £19=£4, 
making  the  answer  £4,  16s.  8Jd. 


Subtract  the  following: 

£      s.      d.        £       s.      d. 

2.  17     10     8i—  14       53:=:? 

3.  119      7     6  —  17     19     5J=? 

4.  500       0     0  —  20     18     8  =? 

5.  176     14    7]—  129     15     7%=? 

Total,  £630,  13s,  9Jd 
116.   To  multiply  compound  numbers.* 

6.  Multiply  £17,  4s,  9J  by  8. 

OPERATION.     £17    4    9| 


£137  18     2 

After   performing    operations    in    addition,    the    learner 
will  readily  see  how  this  is  done. 

7.  £  17  18  8J  X  7==?          10.  £  48     9  6JX2  and  3=? 

8.  £120  16  61X12=?  11.  £145     8 

9.  £365     0  7|X  9=?          12.  £705  13 

Total,  £4860,  15s,  OJd.  Total,  £5639,  19,  6. 

*  Multiplication  may  likewise  be  performed  by  reducing  the  com- 
pound number  to  one  denomination.     (See  Reduction) 
11 


122          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

117.    When  the  multiplier  exceeds  12  and  is  a  composite 
number,  or  otherwise. 
13.  £48,  9, 


£      s.      d.         EXPLANATION. — Here   we  multiply  by    the   two 
48      9      6J    factors  of  which  the  24  is  composed. 
3 


145     8     7^=3  times  the  amount. 


1163     9     0  =8  times  3  or  24  times  the  amount. 

14.  Multiply  £705,  13s,  9|d  by  38. 
£705     13     9|X2 


4234       2  10|=6  times  the  amount. 
6 


25404     17     3  =6  times  6  or  36  times  the  amount. 
1411       7     7^=2  times  the  amount. 

26816       4  lQ$=Sum  0/2+36  times  or  3%Xthe  amount. 

15.  £19     6s  7  dX  84=?          18.  £27     8s  8  dX  87=? 

16.  £91  18s  5idX  89=?          19.  £77  17s  7|dX   95=? 

17.  4s  7JdXl29=?    .      20.  £89  17s  6  dx!50=? 

21.  £176  Xl7=?  24.  £5  7  6|X26=? 

22.  £349  X19=?  25.  £638  X29=? 

23.  £4  5  7^X23=?  26.  £8  4  7|X30=? 

Answers:  £23266,  15,  4|;  £9834,  12,  5;  £566,  1,  3; 
£183,  7,  11. 

118.   To  divide  compound  numbers. 

British  money  being  almost  the  only  thing  in  business  to  which 
compound  numbers  is  applied  exercises  in  it  have  received  most  at- 
tention; and  especially  as  direct  importation  gives  the  clerk  more 
to  do  with  it  than  heretofore.  See  index  for  shorter  methods  of 
computing  this  kind  of  money 


COMPOUND  NUMBERS.  123 

27.  Divide  £157,  13,  6J,  by  5. 

EXPLANATION.— 5  is   contained   in  £157  31 

£         s.       d.      times  and  £2  over.     These  £2  reduced  to  shil- 

5)157      13      6J-      lings,  and  added  to  the  13s.  of  the  dividend, 

31      10      8^-      make  53s.,  in  which  5  is   contained   10  times 

and  3s.  left.     In  3s.  there  are  36  pence,  which, 

added  to  the  6d.  of  the  dividend,  make  42d.,  in  which  5  is  contained 
8  times  and  2d.  over.  In  2d.  and  |d.  there  are  10  farthings,  in 
which  5  is  contained  2  times,  making  j  or  ^d. 

28.  Divide  £157,  13s,  6Jd  equally  between  25  persons, 

25)  £157,  13s,  6Jd(£6,  6s,  IJd,  or  £6,  6s,  l|d,  nearly. 
150 

£7  =remainder. 
20 


W3=shilUngs  in  £7,  with  13s  of  the  dividend  added. 
JL50 

3=remainder  in  shillings. 
12 


42=pence  in  3  shillings  and  6  pence  from  the  div  d. 


Yl=remainder  in  pence. 
4 


70=/ar things  in  17  pence  and  \. 
50 


20=remainder)  or  |-5=-f  farthings. 

£     s.     d.  £      s.    d. 

29.  487  13     0  -r-  9=?  32.  167  18  6|-5-  25=? 

30.  356  7  10  -r-36=?      '  33.  768  14  3 J -5-125=? 
*31.  419  15  6£-r-14=?       34.  17  11  3|-r-875=? 

Answers  :  £12,  17,  8|,  and  £94, 1,  4|. 

*  When  the  remainder  from  farthings  is  J  or  over,  add  a  farthing, 
otherwise  omit  it* 


124          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


GENERAL  EXERCISES  IN  COMPOUND  NUMBERS. 

28.  In  4  yards  2  feet  seven  inches,  linear  measure,  how 
many  inches? 

29.  In  100  inches  how  many  yards? 

30.  Reduce  3520  yards  to  miles. 

31.  In  100,000  inches  how  many  miles? 

32.  How  many  times  will  a    carriage-wheel  turn   in   a 
distance  of  17  miles,  the  wheel  measuring  2  yards  2  feet 
in  circumference? 

33.  From  a  plank  17  yards  long  was  cut  10  yards  2 
feet  3  inches;  how  much  of  it  was  left? 

34.  Reduce  3  acres  140  rods  to  square  yards. 

35.  Divide  200  acres  100  rods  into  10  equal  parts. 

36.  From  406  acres  17  rods  take  68  acres  148  rods  15 
yards. 

37.  Divide  64  acres  134  rods  8  yards  into  5  equal  parts. 

38.  Find  the  price  of  9|  ounces  of  gold  at  £3  17s  8d 
per  Ib.  Troy.* 

39.  Find  how  often  £2  4s  6d  is  contained  in  £41  10s 
7>d. 

40.  Reduce  1  Ib.  3  oz.  5  pwt.  to  pennyweights. 
Answers:  175;  11220;  1,  4,  24,  5,  2,  4;  2;  2,  2,  4;  6, 

9;  337,  28, 15}  ;  2,  19,  10^;  12,  3,  34,  25£;  18|||;  18755; 
20,  10 ;  305. 

The  Teacher  should  require  his  scholars  to  give  the  denomination 
of  each  item  in  the  answers. 

*Find  the  price  of  1  ounce  j  then  the  price  of  9J. 


SHORT  METHODS  OF  MULTIPLYING.  125 


XIII.  SHORT  METHODS  OF  MULTIPLYING. 

119.  BESIDES  the  contractions  by  aliquots,  under  Art. 
66,  the  expert  accountant  and  arithmetician  can  find  ab- 
breviated methods  adapted  to  almost  every  calculation.     A 
few  will  be  given  in  this  place,  to  admit  of  the  learner  using 
them,  when  opportunity  OCCIMTS,  in  the  subsequent  exercises. 

120.  To  multiply  by  11,  write   the  first   figure  of  the 
multiplicand   as   the   first   of  the   product,    and   add  each 
figure  on  the  left  to  the  one  on  the  right,  as  below. 

1.  38397X11=427867. 

Prove  the  following  by  multiplying  in  the  ordinary  way  : 

2.  379X11=?  7.  $219.168X11=? 

3.  1487XH=?  8.  $716.573X11=? 

4.  $37.486X11=?  9.  $316.144x11=? 

5.  $9314.20  XH=?  0.  $137.211X11=? 

6.  $167.473X11=?  11.  $710.22   XH=? 

121.  To  multiply   by  the   teens  when  the  tables  are  not 
known,  and  by  such  numbers  of  two  figures  as  end  with  1, 
as  21,  31,  etc.,  multiplication  by  the  figure  1  ought  to  be 
omitted. 

12.  Multiply  3174  by  17. 

3174  EXPLANATION.  —  The  product  of  7  is  written  one  place 

22218        to  the  right  to  allow  the  first  line  to  stand  in  the  tens' 
place,  by  which  it  is  multiplied  by  10. 


13.  $3163        X15=  ?  18.     $435.16fXlG=? 

14.  $216.37   X19=?  19.     $213.14  Xl8=? 

15.  $1139.24  X13=?  20.  $1137.37^X19=? 

16.  $413.22   X18=?  21.     $713.1Hxlti=? 

17.  *8131.18|X14=?  22-  $4302.87  X  17=? 


*Call  the  J  of  a  cent  25  hundredths,  making  1311825   for  the 
multiplicand,  and  point  oft'  four  figures.     (See  note,  page  06.) 


126 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


23.  $316.27X51=? 

31627  EXPLANATION. — The  5  of  the  multiplier  being  in  the 

158135          tens'  place,  the  first  figure  of  the  product  -is  written 
if  19 077        under  the  tens  of  the  multiplicand. 


24.  $137.50  X31=? 

25.  $298.67  X&1=? 

26.  $783.37JX61=? 

27.  $313.17   X81=? 

28.  $1987.871X91=? 


29.  $2136.22  X  71=? 

30.  $1394.311  X  41=? 

31.  $653.18|X  21=? 

32.  $291.16*  Xl21=? 

33.  $312.18|X  21=? 
122.    When  the  multiplier  wants  from  1  to  12  of  being 

100,  200,  3000,  etc.,  the  work  may  be  contracted  by  mul- 
tiplying by  one  of  these,  and  subtracting  as  many  times 
the  multiplicand  as  the  multiplier  is  short  of  it. 
34.  To  multiply  424  by  97. 

OPERATION.     424x100=42400 
424X     3=  1272 


35.     765X192=? 

30.  1789X398=? 
37.  67.84X188=? 
9876X191=-? 
671 X  B9=? 
59X689=? 


38. 
39. 
40. 


41128 

$89 
$167 

$37.98 
$478.96 


X784=? 
X  29=? 
X489=? 
X499=? 


41. 
42. 
43. 
44. 

45.  $674.82^X992=?' 

46.  $7164.37-^X  87=? 

123.  When  the  multiplier  is  29,  39,  49,  etc.,  we  multi- 
ply by  the  next  higher  number  and  subtract  the  multipli- 
cand, 

47.  To  multiply  176  by  59. 

OPERATION.     176X60=10560 
J  176       . 

10384 

124.  To  multiply  ly  601,  1003,  90001,  etc. 

The  case   differs   from  Art.  8   only  in   the  intervening 


SHORT  METHODS  OF  MULTIPLYING  127 

figures;  so  the  product  is  written  one  place  further  to  the 
right  or  left  for  every  cipher. 

^48.  Multiply  317  by  601.  OPERATION.         317 

1902 

Ans.  190517 

49.  Multiply  15704  by  10007      OPERATION.  15704 

109928 

Ans.  157149928 

125.  When  one  part  of  the  multiplier  contains  the  other 
without  a  remainder,  as  248.  Here  24  contains  3  times 
the  8  or  first  figure,  so  by  multiplying  the  product  of  8 
times  the  multiplicand  by  3,  one  line  is  saved. 

50.  Multiply  76439  by  248. 
OPERATION.        76439 

611512=8  times  76439 
1834536_=3     "     611512 

18956872  Ans. 

REMARK. — This  operation  might  be  shortened  by  multiplying  the 
product  by  8  mentally,  and  adding  that  line  for  the  whole  product. 

51.  Multiply  25938  by  936. 

OPERATION.  25938x936  or  _  25938 

233442  ~~       233442.. 
933768 


24277968 


24277968 

52.  11457X324=?  57.     7832x64256=? 

53.  672x189=?  58.     7498x16144=? 

54.  783x357=?  59.     9739X  3972=? 

55.  924X218=?  60.     6487X  8109=? 

56.  596X426=?  61.  74675X  7206=? 

126.  To  multiply  l>y  375,  625,  750  or  875,  we  first  mul- 
tiply by  125  (Art.  66),  and  that  product  by  3,  5,  6  or  7, 
these  numbers,  375,  etc.,  being  multiples  of  that  number. 


128          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

1649000X5     8245000 

62.  1649X625=—     — g^-= g —1030625 

63.  3156X375=?    $1703.20x750=?    $1456x875=? 

127.  To  multiply   by  a  composite   number   composed  of 
factors  under  13,  the  latter  may  be  used  as  multipliers 
instead  of  the  former. 

64.  314X72. 

OPERATION.     314x12=3768x6=22608. 

65.  932X64=?  68.     $913.27  X^4=? 

66.  738X48=?  69.     $293.75   X72=? 

67.  426X96=?  70.  $6318.371x63=? 

128.  To  multiply  by  any  number  of  9«s,  we  multiply  by 
the  next  highest  number  and  subtract  the  multiplicand. 

71.  3145X999- 

3145X1000=3145000 
Prom  which  take       3145 

3141855 

129.  To   square,  mentally,   numbers  under  39  that  end 
with  9. 

72.  What  is  the  square  of  29? 

29          EXPLANATION. — Writing  1  for  the  first  figure  of  the  pro- 
29      duct,  we  add  1  to  the  tens'  place  of  the  multiplier,  and  rnul- 
7777      tiply  the  sum  on  the  multiplicand  less  1 :  3X28=84,  with 
the  1  annexed=841. 

73.  Find  the  square  of  the  following  numbers  mentally: 
99,  59,  119,  79,  19,  69,  129,  89. 

130.  To  square  any  number  of  9s  instantaneously,  and 
without  multiplying. 

Commencing  at  the  left,  we  write  as  many  9s,  less  one, 
as  the  number  to  be  squared,  an  8,  as  many  Os  as  9s  and 
a  1. 

74.  The  square  of  9999999  is  99999980000001. 


SHORT  METHODS  OF  DIVIDING.  129 

The  square  of  any  number  of  3s  will  be  one-ninth  of 
the  square  of  the  9s. 

131.  To  square  numbers  under  135  ending  with  5. 

The  first  two  figures  on  the  right  of  the  product  will 
always  be  25 ;  and  to  find  the  others,  we  add  1  to  the 
tens'  place  and  multiply  it  on  the  tens'  and  hundreds' 
places  above. 

75.  To  square  115.  OPERATION.     11 

12 

13225 

The  reason  of  this  will  be  apparent  by  multiplying  in 
the  usual  way. 

132.  To  square  a  number  containing  a  half,  as  12J,  we 
multiply  the  whole  number   by  the    next   higher    number 
and  add  a  fourth.     8|  squared=8x9-|-{=72J. 

76.  Find  the  square  of  the  following  numbers:  99999, 
33333,  75,  45,  65,  62£  16J,  19J. 


XIV.  SHORT  METHODS  OF  DIVIDING* 

133.   Division  may  often  be  contracted  by  cancellation* 
tvhen  the  terms  are  written  in  fractional  form. 

1.  Divide  1463  by  28. 

209  EXPLANATION.—  The  terms  1463  and  28   were   first 

divided  by  7,  leaving  209  fourths,  and  209  divided  by 

*   4  Sives  52i- 


To  THE  TEACHER.  —  The  author  does  not  offer  all  these  contrac- 
tions as  rules  of  general  utility;  still,  he  is  of  opinion  that  a  fa- 
miliar knowledge  of  them  will  be  advantageous  to  the  student  of 
arithmetic  in  disciplining  his  mind  and  showing  him  the  relation  of 
numbers.  Where  the  instructor  thinks  otherwise,  he  can  omit  them. 

*  To  cancel  signifies  to  blot  out  or  make  void. 


130          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Prove   the  answers   obtained,  by  dividing  in    the  usual 
way. 

2.  3465-v-35=?     2763-4-  81—?  $65.45--243=  ? 

3.  1962-r-22=?     6876-r-152=?  $54.36-r-144=z? 


134.  To  divide  by  aliquots  of  100,  1000,  etc. 
This  process  is  the  reverse  of  that  under  Art.  66. 

135.  To  divide  by  a  composite  number,  as  96,  which  is 
composed  of  the  factors  12  and  8,  or  648,  which  is  com- 
posed of  9X8X9.     This  operation  is  performed  by  using 
the  factors  instead  of  the  whole  number. 

4.  Divide  $78.54  by  32.  OPERATION.     4)7854 

8)1963—2 

245—3 

How  the  true  remainder  is  found: 

The  first  remainder  is  2  cents,  because  it  is  left  from  the 
cents  that  were  divided.  The  second  remainder  is  four 
times  as  great  as  if  it  were  from  the  first  line,  because 
every  figure  of  the  second  line  is  four  times  as  great  as  if 
it  stood  in  the  first  line.  Four  times  3—12  and  the  2  of 
the  first  remainder  equal  14,  the  true  remainder. 

Ans.  245£f  . 

5.  Divide  6371  by  336.  OPERATION.     6)6371 

7)1061—5 

REMARK.  —  The  true  remainder  of  this  example  is  -  —  ~~ 
found  by  multiplying  the  last  remainder  by  7,  to  '_l__ 
make  it  of  the  same  value  as  if  it  were  from  the  line  18  —  7 

above,  and  that  by  6,  to  make  it  of  the  same  value  as  if  it  were 
from  the  upper  line:  7X?XG:=r:2H  to  which  add  6X4+5  or  29. 
The  true  remainder  is  323.  Ans.  1  8|||. 

6.  Divide  1463  by    28  10.  4571-f-441==? 

7.  "       7614   "     72  11.  1987-^379=? 

8.  "       1943   "      49  12.  9843-^-720==? 

9.  "       8765    "   343  13.  1456-f-729=? 


SHORT  METHODS  OF  DIVIDING.  131 


14.  Divide  7654  by  25. 


76.54=10<M  of  7654. 
4 


306.16=4  times  as  much,  or  ^  of  7654. 
or  306^=306^. 

15.  $3675-^-125  =  ?    S213.67-s-16£=?    $11  74-4-121=? 

16.  $2153--  33$=?    $319.25---  8$=?        316-^25  =? 

136.  When  the  divisor  is  15,  35,  45,  55  or  65,  it  will 
abbreviate  the  work  to  multiply  the  dividend  by  2  and  di- 
vide by  30,  70,  etc. 

17.  345-^-35=?  345x2=690-^-7=98f 

18.  2756-4-15=?     $1324.25-4-35—?     $365.75-4-45=? 

137.  To  divide   by  75,  175,  225   or  275,  the  dividend 
may  be  multiplied  by  4  and  the  product  divided  by  300, 
700,  900  or  1100. 

19.  2136-4-75=? 

2136          EXPLANATION.  —  4  three-hundred  ths  being  equal 
4      to  1  seventy-fifth,  it  abbreviates  the  work  to  divide 
3  00^)85  44     b^  ^^  an(*  mult*P1y  by  4. 


20.  3678-4-175=?     6317-r-175=?     $19.32-f-275=? 

138.  Long  division  may  be  abbreviated  by  performing  a 
part  of  the  process  mentally,  and  writing  only  the  result. 

21.  76354-^-34=? 

34)76354(2245      EXPLANATION.  —  The  products  are  omitted;  only 
83  the  remainder  and  the  figures  brought  down  are 

155  written. 

194 
24 

22.  3167^-184=?     1679-i-21=?     67831-4-498=? 

139.  To  divide  by  any  number  of  9s,   the  dividend  may 
be  pointed  off,  from  the  left,  in  periods  corresponding  with 


132          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

the  number  of  figures  in  the  divisor.  If  the  divisor  have 
two  9s,  it  may  be  pointed  off  in  periods  of  two;  if  three 
9s,  in  periods  of  three,  and  the  operation  performed  as  in 
the  formula. 

23.  316328-;-99^?     7312961--999=? 

31,63,28  731,296,1  REMARK.— It  will  be  observed  that 

31,63  731,2  each  line  in  the  formula  is  diminished 

31  7  by  one  period,  corresponding  with  the 

01  OR  oo  ^7OOA  OQA  number  of  9s,  and  the  sum  of  the  whole 

oiyo.-u.iu          lo^u.^oU       ,  .  .,  p.     ,     , . 

1  -.         taken.     As  many  figures  as  Us  should 

I ~ be  pointed  off  for  a  remainder,  plus 

3195.23  7320.281        the  carrying  figure  of  the  first  line. 

24.  Divide  167389  by  9999  and  7654321  by  99.* 


XV.  MARKING  GOODS— GAIN  AND  LOSS. 

140.  ON    receiving    goods    for    sale,    the    merchant    or 
some  of  his  clerks 'examine  them  by  the  bill  or  invoice, 
•which   is   usually  received   in    advance   of  them  by  mail, 
after  which  they  proceed  to  mark  them. 

141.  Marking   Goods. — This  is  done  by  selecting  sam- 
ples of  each   kind   or   quality   of  goods,   and   putting   on 
them  a  private  mark,  indicating  the  cost  price,  the  selling 
price,  or  both. 

Every  house  has  its  own  private  mark,  which  usually 
consists  of  a  word  or  phrase  to  represent  the  ten  digits, 
as  the  word  importance,  which  has  ten  letters.  Commenc- 
ing at  1,  the  letters  are  arranged  as  follows: 

importance 
12          34567890 
In  addition  to  these,  another  letter  not  contained  in  the 
above  is  selected  for  a  repeater,  so  that  when  a  figure  oc- 

*  It  would  be  no  abbreviation  to  divide  by  9  in  this  way. 


MARKING  GOODS— GAIN  AND  LOSS.  133 

curs  twice  it  may  be  inserted  to  prevent  detection.  The 
letter  g  will  do  in  this  case. 

To  mark  *6.55,  the  letters  trg  would  be  used,  with  per- 
haps a  line  or  period  to  separate  the  dollars  from  the  cents. 

To  fix  on  a  selling  price,  various  circumstances  have  to 
be  taken  into  consideration-^-the  cost  of  transportation, 
the  probable  length  of  time  required  to  sell,  cost  of  rent, 
wages  of  clerks,  depreciation  of  stock,  etc. 

A  certain  rate  is  put  on  for  the  first  item,  say  5  to  15 
per  cent.;  then  an  amount  estimated  to  cover  the  balance 
and  leave  a  profit.  When  adding  the  entire  rate  of  ad- 
vance to  the  cost  price,  the  clerk  is  not  required  to  be 
exact,  as  simplicity  of  calculation  is  a  greater  object  than 
uniformity  of  profit.  Thus,  in  marking  goods  to  sell  by 
the  dozen,  a  multiple  of  12  would  be  preferred,  whether  a 
little  above  or  below  the  fixed  rate.  For  the  same  reason, 
aliquots,  or  numbers  formed  of  aliquots,  of  100  would  be 
selected. 

When  working  the  following  exercises,  the  learner  will 
remember  that  fractions,  unless  occurring  with  aliquots  of 
100,  are  usually  omitted.  When  under  ^  they  are  rejected; 
otherwise,  a  cent  is  added  to  the  cents/* 

1.  Add  25%  to  $3.50,  $5.75,  $1.82,  .75,  $2. 
Answers:  $4.372,  $7.18f,  $2.28,  .94,  $2.50. 

2.  Add  16|- %  to  $6.20,  $3.122,  27c,  $3.87. 

3.  Add  37£%  to  20c,  122c,  $1.15,  6c,  15c. 

4.  Add  15%  to  $1.20,  75c,  $1.22,  $3.57,  162c,  27c. 

5.  Add  33J%  to  5c,  15c,  21c,  $6.75,  18c,  $27.50. 
Answers:  $15.72,  $8.24,  $46.46,  $2.32. 

*To  indicate  the  degree  of  exactness  required,  a  few  answers  will 
be  given  to  the  examples  which  follow. 

One  object  in  giving  these  examples  being  to  exercise  the  judg- 
ment of  the  pupil,  he  will  not  be  required  to  obtain  precisely  the 
same  figures  as  given  in  the  other  answers. 


134          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

6.  $1.75+14%  =  ?  14.  $6.75+26%=? 

7.  .87+12%  =  ?  15.       .07+40%  =  ? 

8.  $1.67+53%  =  ?  16.  $3.60+12%  =  ? 

9.  .04+60%  =  ,?  17.  $4.80+30%  =  ? 

10.  $2.75+35%  =  ?  18.  $1.20+35%  =  ? 

11.  .16+15%  =  ?  19.  $7.20+40%  =  ? 

12.  .05+25%=?  20.  $6.00+14%  =  ? 

13.  $3.14+18%=?  21.  $5.10+25%  =  ? 
Answers:  $13.25,  $43.79. 

22.  To       .30  add  20  %  profit. 

23.  To  $1.20  add     5   %  charges  and  20%  profit. 

24.  To  $1.75  add     2J%  freight  and  10%  profit. 

25.  To       .08  add  16| %  profit. 

26.  To       .36  add  33J %  profit. 
Answer:  $4.41. 

27.  $0.10+10%=?  30.  $1.00+16|%=? 

28.  .05+50%=?  31.  $3.25+334%  =  ? 

29.  .25+20%=?  32.  $1.40+12^%=? 
Total,  $7.57. 

The  following  may  be  worked  by  aliquots  of  100: 

33.  Add    5   %  to  $12.50         44.  To  $3.50  add    4  % 

34.  "  6|-%  "  $7.80  45.  "   $9.50    " 

35.  "  84%  «  $4.85  46.  «   $0.10    " 

36.  "  9  %  "  $3.50  47.  "   $0.18f " 

37.  "  124%  "  $0.87  48.  "   $0.06^  "    40  % 

38.  "  20  %  "  $0.45  49.  "   $0.87^-  "    50  % 

39.  "  18f%  «  $0.15  50.  "   $0.084  «    75  % 

40.  "  25  %  «  $0.124  51.  "   $0.11J"    20  % 

41.  "  7  %  "  $6.20  52.  "   $0.80    " 

42.  «  34%  "  $1.00  53.  "   $0.16    " 

43.  "  12J>%  "  $1'.35  54.  ««   $0.47    « 
Answers:  41.53,  $18.56. 


GOODS— GAIN  AND  LOSS. 


135 


ORAL  EXERCISES .* 


1.  $0.10+  25  %  =  ? 

18.  $0.50  +  20  %  =  ? 

2.       .25+  10  %  =  ? 

19.       .25  +  50  %  =  ? 

3.       .20+     6   %  =  ? 

20.       .11  +  15  %  =  ? 

4.       .33+  33J%  =  ? 

21.       .12^+  20  %  =  ? 

5.       .16+  25  %  =  ? 

22.       .45  +     9  %  =  ? 

6.       .45+     5  %  =  ? 

23.       .66  +  33|%  =  ? 

7.       .87+  30  %  =  ? 

24.       .14  +     5  %  =  ? 

8.    .16+  m%=? 

25.       .35  +  20  %  =  ? 

9.       .10+  22  %=*\ 

26.       .16  +  50  %  =  ? 

10.       .05+  30  %  =  ? 

27.       .67  +  12  %  =  ? 

11.  $2.20+  50  %  =  ? 

28.  $1.20  +  16|%  =  ? 

12.  $550  +  100  %=? 

29.       .27  +  50  %  =  ? 

13.       .75+  60  %  =  ? 

30.  $6.50  +  16  %  =  ? 

14.  $1.25+  30  %  =  ? 

31.  $2.15  +     8  %  =  ? 

15.  $1.12+  25  %  =  ? 

32.  $1.87  +200  %  =  ? 

16.       .75+  20  *T=? 

33.       .31  +  18  %  =  ? 

17.     1.50+     8  %  =  ? 

34.       .19  +  12£%  =  ? 

35.  30%  of      $20.00=? 

43.     $0.37^+30  %  =  ? 

36.  50%  of  $1000.00==? 

44.         .62J+  5  %  =  ? 

37.  25%  of    $700.00  =  ? 

45.     $1.25  +12J%  =  ? 

38.  16%  of    $500.00=? 

46.         .16  +  5  %  =  ? 

39.  10%  of    $350.25  =  ? 

47.         .54  +40  %  =  ? 

40.  15%  of    $200.00=? 

48.  $30.00  +60  %  =  ? 

41.  80%  of    $500.00=? 

49.  $90.00  +  9  %  =  ? 

42.  20%  of      $20.00=? 

50.  $70.00  +30  %  =  ? 

*  The  Teacher  may  require  the  flections  ill  these  exercises. 


136          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


GAIN  AND  LOSS. 

142.  Merchants  distinguish  between   real   gain  or  loss 
and  gain  or  loss  per  cent.,  calling  the  former  the  actual 
gain  or  loss,  and  the  latter  the  gain  or  loss  per  cent. 

143.  To  find  the  actual  gain,  it  is  simply  necessary  to 
subtract  the  cost  price  from  the  selling  price. 

1.  Bought  a  house  and  lot  for  $4367  and  sold  them  for 
$5000;  how  much  did  I  gain? 


=(7os£  or  buying  price. 
$633=:-4c2waZ  gain. 

Cost  price.  Selling  price.  Cost  price.  Selling  price. 

2.  $2.75  $4.87  7.  $316.17        $215.25 

3.  $97.35         $120.10  8.  $112.14        $120.48 

4.  $6.87  $6.98  9.  $317.18|      $21.9.m 

5.  $5.40  $9.80  10.     $67.21          *8&2J8 

6.  $3.20            $6.40  11.     $54.12J        821.18J 
Total  gain,  $32.58.     Net  loss,  or  loss  with  gains  de- 

ducted, $204.51. 

144.  To  find  the  gain  per  cent.,  is  simply  to  find  the 
gain  on  every  hundred  dollars  or  cents. 

Required  the  gain  per  cent,  on  goods  which  sold  at 
$1.35  and  cost  $1.20. 

135—120=15  cents,  the  actual  gain  on  120.  15-=- 
120=Tyg=gain  on  1  cent.  y'&X  100=^^=12  J  per 
cent.,  or  gain,  on  100  cen-ts. 

OPERATION.     135  EXPLANATION.  —  The  actual  gain 

120  is  first  found;  then  the  gain  per 

cent.,  by  dividing  the  actual  gain 
(when  multiplied  by  100)  by  ti& 
first  cost. 


MARKING  GOODS—  GAIN  AND  LOSS.  137 

12.  Goods  which  cost  $2.00  were  sold  at  $3.00;  required 
the  gain  per  cent. 

13.  The  cost  price  was  $1.25;  the  selling  price,  $1.50; 
what  was  the  gain  per  cent.? 

14.  Goods  bought  at  75  cents  sold  at  $1.00;  what  was 
the  gain  per  cent.  ? 

15.  10   cents  was  the   cost;  12  \  the   selling  price;  re- 
quired the  gain  per  cent. 

Total  rates  of  gain  of  the  four, 


First  cost.        Selling  price.  First  cost.  Soiling  price. 

16.  $12.50   $10.00     20.  $3167.00   $3000.00 

17.  .18     .20     21.  $1000.00   $1500.00 

18.  .05     .06     22.   $27.80    $20.00 

19.  $127.52     $111.58  23.       $12.17          $11.50 
Net  loss,  on  16  to  19,  \-^%.     Answers  to  second  group, 


24.  Bought  a  bbl.  of  apples  for  $1.75  and  sold  it  for 
$2.25;  what  did  I  gain  per  cent.? 

25.  Sold  25  bbls.  of  potatoes  for  $39.00;  how  much  did 
I  gain  per  cent.,  if  they  cost  me  $1.25  per  barrel? 

26.  Bought  150  bbls.  of  flour  @  $5.25,  paid  for  dray- 
age  $7.50  and  porterage  $1.00;  at  what  per  barrel  should 
I  sell  it  to  gain  15  per  cent.? 

27.  Bought  15  horses  at  $125  each,  and  sold  the  lot  for 
$3500;  what  was  my  gain  per  cent.,  after  paying  $25  for 
their  feed? 

28.  Sold  a  safe  which  cost  me  $80  for  $75;  what  was 
my  loss  per  cent.? 

29.  Bought  a  bill  of  goods  for  $350,  paid  freight  $15.20, 
insurance  $5,  drayage  $3,  and  sold  them  for  $425;   what 
was  my  actual  gain,  and  what  my  gain  p^er  cent.? 

30.  Sold  A's  note  for  $750  at  a  discount  of  15%;  what 
did  I  pay  for  it? 

12 


138          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

31.  Sold  B's  note  of  $320  for  $300;  what  was  the  rate 
of  discount? 

Answers:  $637.50,  $51.80,  14%  nearly,  84J%  nearly, 
6i%>  6f%>  28T%>  24f%>  $6-10- 

145.  When  the  selling  price  and  the  rate  per  cent,  are 
known,  to  find  the  first  cost. 

32.  What  was  the  first  cost  of  goods  marked  $2.65,  the 
rate  per  cent,  of  profit  being  25? 

1.25)2.65(212  EXPLANATION.— Every  dollar  invested  in 
2  50        or  $2.12  the  goods  has  increased  25  cents,  and  is 
-                          worth  125  cents.    Hence,  there  are  as  many 
irQfl                        invested  dollars  in  $2.65  as  1.25  is  con- 
tained  times  in  it.     The  remainder,  15  dol- 
lars, we  reduced  to  cents,  which,  divided  by  1.25,  gives  12  cents. 

Another  way:  Since  25%  is  J  of  100,  the  $2.65  must 
be  I  more  than  the  first  cost.  Let  the  first  cost  be  f, 
then  J+|=|.  Therefore,  $2.65=f  of  the  first  cost. 

2|5.=53=one-fourth  of  the  first  cost. 
53X4=2.12,  first  cost. 

What  was  the  first  cost  of  the  goods  marked  as  follows? 
The  learner  can  prove  his  calculations  by  reversing  the 
process. 

33.  $2.25  @  10  %  gain       38.  $2.87    @  10  %  loss 

34.  $3.70  "    5  %     "          39.  $1.54    "     6  %     " 

35.  $115.87  "  12%%     "          40.  $3.75    "  25  %     " 

36.  $14.54  "  ^1%     "          41.       .87J  "  12%%    « 

37.  .87  "  16f%     "          42.       .12%  "  5o"'%    " 

43.  $9.50  @  50  %  gain  47.  $90.00  @  20  %  gain 

44.  $7.87  "  25  %     "  48.  $75.30  "  15  %     « 

45.  $6.50  "  16f%     "  49.  $82.50  ' 

46.  $8.75  «  18f  %     "  50.  $60.00  «  20 


COMMISSION  AND  BROKERAGE.  139 


XVI.  COMMISSION  AKD  BROKERAGE* 

146.  COMMISSION,    or    brokerage,    is    the    percentage 
charged  by  a  commission  merchant,  factor,  agent  or  broker 
for  transacting  business  for  another. 

147.  Commission    is    usually    reckoned    on    the    whole 
amount  of  sale,  purchase  or  collection. 

1.  At  2J  per  cent.,  what  is  the  commission  on  $17640? 

Ans.  $441. 

2.  A  merchant  sells  goods  for  another   to   the   amount 
of  $4371.81;  what  is  his  commission  at  5  per  cent.? 

3.  A  broker  receives  |  per  cent,  for  selling  $2500  worth 
of  merchandise  for  a  commission  merchant;   what  is  the 
amount  of  his  brokerage? 

4.  A  of  New  Orleans  buys  sugar  for  B  of  Cincinnati  to 
the  amount  of  $7100;  what  is  the  amount  of  his  commis- 
sion at  1J  per  cent.? 

5.  A  commission  merchant  sells  goods  for  his  principal 
to  the  amount  of  $3000,  and  charges  2^  per  cent,  commis- 
sion; what  does  he  make  by  the  operation,  after  paying  a 
broker  |  per  cent,  for  his  services  in  effecting  sales? 

6.  After    receiving    5    per    cent,    commission    on    sales 
amounting  to  $520.75,  how  much  should  I  return  to  my 
principal? 

7.  Gave  a  lawyer  a  note  of  $50  to  collect,  at  8  per  cent.  ; 
how  much  should  I  receive? 

Omit  fractions  of  a  cent  in  the  answer: 

Answers:  $106.50,  $6.25,  $218.59,  $67.50,  $494.71,  $46, 
$510.71. 

• ; 

*See  Commission  Merchant,  page  94;  Brokers,  page  95. 


140          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

What  is  the  commission  on  the  following  amounts? 

8.  $364.15  @  3£$i  =  ?  12.       §36.21  @ 

9.  $78.54  «  6^%  =  ?  13.     $174.09 

10.  $710.06  «  83%  =  ?          14.  $2167.90 

11.  $876.75  "  9J%r=?  15.       $78.21  "  $%  =  1 

Total,*$161.31.  Total  $146.13. 

16.  A  commission  merchant  charges  2J%  com.  and  2^r% 
guarantee  on  the  sale  of  goods  amounting  to  $3100;  how 
much  should  he  return  to  his  principal? 

17.  A  merchant  receives  a  consignment  of  goods  valued 
at  $3000,  and  sells  them  for  $4500 ;  how  much  should  he 
remit  to  the  consignor,  after  reserving  4J%  for  com.  and 
guar.  ? 

Answers:  $2945,  $4297.50,  $5297.30. 

18.  A  merchant  sells  a  note  of  $100  to  a  money  broker, 
at  a  discount  of  6  per  cent.;    how  much  money  does  he 
receive? 

19.  A   New   York   merchant   buys  a  bill   of  exchange 
worth  $400,  on  a  Cincinnati  banking  house,  at  J  per  cent, 
discount;  how  much  does  he  pay  for  it? 

20.  Bought  a  bill  of  exchange  on  New  York  for  $7691, 
at  l^r  premium;  what  did  I  pay  for  it? 

21.  Purchased  20   shares   of  railroad  stock,  worth  $20 
per  share,  @  12^%  discount,  and  sold  it  at  par;  what  was 
the  amount  paid?  and  how  much  did  I  gain? 

Answers:  $94,  $399,    $7806.37,  $350,  $50. 

22.  "What  is  the  premium  on  the  following?     $31.46@ 
5J%;  $1760@6i%;  $4617@9i%.  Total,  $550.35. 

148.    To  find  tlie  commission  on  investments. 

The  merchant  often  has  moneys  in  his  hands  or  re* 
mitted  to  him  for  investment  in  goods  or  stocks,  upon 
which  he  is  allowed  commission  on  the  amount  invested 
only. 


COMMISSION  AND  BROKERAGE.  HI 

23.  At  2-|-%  commission,  what  amount  of  money  shall  I 
retain  of  $2000  in  my  hands  for  investment? 

This  $2000  contains  100%  of  the  amount  to  be  invested, 
plus  2.5%,  my  commission  making  102,5%  of  it. 

Reducing  both  to  tenths,  we  have  2000.0  to  be  divided 
by  102.5. 
102.5)2000.0(19.5121  or  $19.5122=1%, 

1025  Which,  multiplied  by  2J  gives  the  com- 

9750  mission,  or  by  100,  gives  100  per  cent.,  or 

0995  the  amount  to  be  invested.        Com  $48.78. 

5250000          REMARK.— The  525  dollars  remainder  were 
5125  reduced  to  tenths  of  mills. 

125<T 
1025 

2250 

2050 


2000 
1025 

975 

RECAPITULATION. — To  find  1  per  cent.,  we  divide  2000  by  102.5, 
which,  multiplied  by  100,  gives  100  rer  cent.,  or  the  amount.  The 
difference  is  my  commission. 

24.  What  will  be  my  com.  on  $1300,  to  be  invested  at  2%? 

25.  At  5%  commission,  how  much  sugar  can  I  buy  for 
$3475.25,  when  it  sells  at  20  cents  a  pound? 

26.  Out  of  $987.50  how  many  pounds  of  tobacco  should 
I  purchase  for  my  principal  at  35  cents  a  pound,  commis- 
sion 2|%? 

27.  At  2|-%  commission,  how  much  shall  I  reserve  of 
$2130.67,  after  investing  in  cotton  at  25  cents  a  pound? 

28.  To  pay   12-J-  cents   a  pound  for  rice,  and  -fa%  for 
insurance,  and  myself  3%  commission,  how  many  pounds 
can  I  buy  for  $345.15? 

The  learner  will  prove  his  work  by  reversing  the  process. 


142  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

29.  Invest  $1367.37,  \  in  gloves  at  §30  a  dozen,  \  in 
sheeting  at  35  cents  a  yard  and  the  balance  in  muslin  at 
20  cents  a  yard;  commission  2J%. 

INTEREST. 

149.  Interest  is   a    percentage   allowed   for  the   use  of 
capital,  which  may  consist  of  money,  merchandise  or  debts 
due.     It  is  regulated  by  the  year  or  month. 

150.  The  sum  upon  which  interest  is  reckoned  is  called 
the  principal;  the  percentage  allowed  per  month  or  year, 
the  rate;  and  the  sum  of  the  principal  and   interest,  the 
amount. 

151.  Interest    is    divided    into    simple    and    compound 
Simple  interest  is  percentage  on  the  principal  alone;  com- 
pound interest  is  interest  reckoned  on  both  principal  and 
interest. 

The  legal  rates  of  interest  in  the  several  States  is  vari- 
able. 

152.  Money  loaned   on  interest  is   usually  secured   by 
promissory  notes,  expressed  as  follows: 


153.  This  note  would  be  due  on  the  1st  day  of  Decem- 
ber but  for  a  law  which  allows  three  days  longer  for  its 
payment,  called  days  of  grace.  It  will  therefore  be  legally 
due  on  December  4th,  and  interest  will  be  reckoned  to 
that  date. 

*  Other  forms  of  notes  will  be  found  under  Banking. 


SIMPLE  INTEREST,  143 


XVII.  SIMPLE  INTEREST. 

154.  Interest  is  usually  calculated  on  the  basis  of  360 
days  to  the  year.*     When  notes  are  drawn  by  the  month, 
calendar  months  are  understood.     Thus,  a  note  drawn  on 
the  1st  of  September,  as  the  preceding,  falls  due  on  the 
1st  of  December,  plus  the  days  of  grace. 

155.  The  simplest  method  of  computing  interest  is  to 
do  it  at  the  rate  of  6  per  cent,  per  annum,  and  add   or 
subtract  when  it  is  higher  or  lower  than  that. 

156.  The  interest  for   60   days   at   6%    per  annum   is 
equal  to  as  many  cents  as  there  are  dollars,  or,  in  other 
words,  is  1  per  cent,  of  the  principal. 

The  reason  of  this  is  obvious.  6  per  cent,  per  annum 
is  ^  per  cent,  per  month,  or  1  per  cent,  for  two  months  or 
60  days. 

ORAL  EXERCISES. 


1.  The  interest  for  60  days 
2.     "         "           "    60      " 
3.     "         "          "    60      " 
4.     "         "          "    60      " 
5.     "         "          "    60      " 
6.     "         "          "    60      " 
7.     "         "          "    60      " 
8.     u         "          "    60      " 

nt    (\C/ 

,    cLL    \J  yfl  , 

u  6^ 
"  6&/ 
"  6C/ 
"  §°/ 

"  6^ 

"  «*; 

on      $20      =? 
$50      =? 

"     $100      =? 
$75      =? 
"     $125      =? 
"     $175      =? 
"     $200      =? 
"     $316.50=? 

9.     "         " 

"    60      <( 

"  ^9^ 

"     $715      =? 

10.     "         " 

"    60      " 

"  6^! 

"       $50.50=? 

11.     "         " 

"    60      " 

"  *%> 

"      $215.15=? 

12      u         u 

"    60      " 

"  6%' 

"   $1000      =? 

*In  New  York  interest  is  usually  reckoned  for  the  full  year. 


144          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

13.  The  interest  for  60  days,  at  6%,  on    $976.14=? 

14.  «  «  "  60  "  «  6%,   «  $715.15=? 

15.  «  "  "  60  "  "  6%,    "  $5000      =? 

16.  "  "  "  60  "  "  6%,    "  $5      ==? 

17.  "  "  "  60  "  "  6%,    "  $23.13=? 

18.  «  "  "  60  "  "  6%,   «  $67.15=? 

19.  On      $7.50=?  28.  On    $269.14=? 

20.  "  $8,75=?  29.  "  $198.97=? 

21.  "  $118.67=?  30.  «  $267.18=? 

22.  «  $368.56=?  31.  «  $1365.50=? 

23.  "  $210.33=?  32.  c<  $316.18=? 

24.  "  $67.67=?  33.  "  $215.16=? 

25.  «  $39.37=?  34.  "  $716.16=? 

26.  "  $21.37=?  35.  "  $317.60=? 

27.  "  $116.16=?  36.  "  $167.37=? 

157.  Having  the  interest  for  60  days,  the  interest  for 
any  shorter  time  may  be  found  by 

ALIQUOTS   OF    60. 

30  days=l  12  days=  i  5  day 3=^  2  days=^ 
20  '••  =£  10  •  «  =  J  4  «  =^  1  day  =^ 
15  «  =1  6  «  =TV  3  «  =5V 

When  the  number  is  not  an  aliquot  of  60, 

For    7  take     6  and  1  For  29  take    1    off  30 

"      8     "       6    "  2            «     35     "     30  and    5 

"     14     «     12    <c  2            "     38     "     30,    6  and  2 

"    19     «     15    "  4           «    43     «     30,  12    «    1 

"     26     "     20    "  6            "    45     «     15    oif  60 

"     27     "     15    "  12            «     85   add  20  and    5^ 

*The  Teacher  should  continue  these  exercises  till  his  scholars  are 
familiar  with  all  the  aliquots,  and  the  method  of  resolving  other  . 
numbers  into  aliquots. 


SIMPLE  INTEREST.  145 

1.  Fin.d  the  interest  cm  $500,  for  30  days,  at  6%  per 
annum. 

3Q=£)5.QQ=lHterestfor  60  days. 
2M=Interest  for  30  days. 

2  to  8.  Find  the  interest  on  $200  for  12  days,  15  days, 
20  days,  10  days,  6  days,  3  days,  1  day,  at  6%  per  annum. 

AHS.  $2.23.3. 

9  to  13.  Find  the  interest  on  $160  for  1,  2,  3,  4  and  5 
days,  at  6%  per  annum.  Am.  $0.39.9+. 

14  to  18.  Find  the  interest  on  $240  for  6,  10,  12,  20 
and  30  days,  at  6%  per  annum.  Ans.  $3.12. 

19  to  22.  Find  the  interest  on  $1000  for  1,  10,  12  and 
6  days,  at  6%  per  annum.  Ans.  $4.83.3+. 

23.  Find  the  interest  on  $675,  for  27  days,  at  6%  per 
annum. 

$6.75  =Iutcrest  for  60  days. 

15=1=  1.687 
12=4-  1?5_ 

3.037  or  $3.04. 

Principal.       Time.  Principal.        Time. 

24.  $250  for  20  ds=?  30.  $650  for  35  ds=? 

25.  $567  for  14  ds=?  31.  $980  for  80  ds=? 

26.  $968  for  25  ds=?  32.  $216  for  93  ds^? 

27.  $846  for  33  ds=?  33.  $800  for  67  ds=? 

28.  $610  for  18  ds=?  34.  $915  for  44  ds=? 

29.  $918  for  27  ds==?  35.  $1200  for  93  ds— ? 
Answers:  $16.803+  and  $54.448+. 

158.  Merchants  or  bankers  seldom  reckon  interest  on 
cents.  When  under  50  they  are  rejected;  otherwise,  a 
dollar  is  added  to  the  dollars. 

It  should  also  be  observed  that  business  men  express 
their  results  in  dollars  and  cents,  to  which  usage  the 
learner  ought  to  conform.  For  practice  or  review,  the 
13 


146 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


51. 


teacher  may  require   the  exact  answers   instead  of  those 
given. 

40.  $1799.14  for  93  ds=? 
$3$7.66  for  67  ds=? 
$199.44  for  41  ds=^? 
$450.22  for  29  ds^? 
Total,  $35.75. 

48.  $1997.00  for    13  ds=? 

49.  $7.88  for    54  dsr^? 

50.  $17.97  for    35  ds=? 
$10.00  for  120  ds^? 

Total,  $4.71. 

58.  $1999.20  for  23  ds=? 

59.  $361.74  for  18  ds=? 

60.  $78.93  for  23  ds=? 

61.  $1467.20  for  34  ds=? 

62.  $7100.18  for  77  ds=? 

63.  $29.00  for  99  ds=? 

Total,  $108.96. 

159.  To  compute  6%  interest  for  any  number  of  months. 

When  used  in  drawing  notes  or  drafts,  the  month  is  al- 
ways calendar;  but  when  computing  interest,  30  days  is 
considered  a  month;  hence,  the  note  on  page  142  would 
draw  interest  for  one  day  more  than  the  three  months, 
exclusive  of  the  days  of  grace,  there  being  94  days  be- 
tween the  date  and  the  maturity.* 

This  is  a  very  important  distinction,  as  will  be  seen  by 
reference  to  Bank  Discount. 

160.  Since  there  are  half  as  many  60  days  as  months, 
we  multiply  the  interest  for  60  days  by  half  the  number 
of  months. 


36.  $1000.00  for  27  ds=? 

37.  $71.97  for  47  ds^=? 

38.  $61.80  for  45  ds^? 

39.  $190.27  for  16  ds=? 

Total,  $6.04. 

44.  $719.99  for  11  ds=? 

45.  $55.18  for    9  ds=? 

46.  $88.17  for  69  ds^? 

47.  $466.00  for  78  ds^? 

Total,  $8.47. 

52.  $1000.00  for    97  ds=? 

53.  $650.00  for    67  ds=? 

$10.70  for  13  ds=? 
$127.57  for  51  ds=? 
$368.17  for  118  ds^? 
$718.57  for  125  ds=? 
-Total.  $46.76. 


54. 
55. 
56. 
57. 


*ln  practice,  when  the  note  remains  unpaid  till  maturity,  interest 
would  be  charged  for  only  the  three  mouths,  as  if  it  were  90  days. 


SIMPLE  INTEREST.  147 

64.  Find  the  interest  on  $620,  for  4  months,  at  6%  per 
annum. 

Q.^= Interest  for  60  days. 

2— Number  of  60  days  in  4  months. 

\2AQ=Interest  for  4  months. 
Find  the  interest,  at  6%  per  annum,  on  the  following: 

65.  $750.25    for  6  mos=:?       69.  §910.70    for  5  mos=? 

66.  $218.87^  for  8  inos=?       70.  $876.33^  for  4  inos=? 

67.  $1000.00^  for  7  mos=?       71.  $937.79     for  3  mos=? 

68.  $560.374  for  9  mos=?       72.  $168.00    for  1  ino  ==? 
Answers:  $91.46  and  $55.21. 

73.  Required  the  interest  on  $350.25  for  7  mos.  15  ds* 

$3.50  =Interest  for  60  days. 

&k=litumber  of  60  days  in  7  months. 

10.50 
1.75 
875=Interest  for  15  days  or  J  of  60. 

$13.125"  or  $13.122. 
Another  way: 

$3.50  ^Interest  for  60  days. 


14.00  =  Interest  for  8  months. 
87 '5= Interest  for  15  days  off. 

13.125  or  $13.12*. 
Compute  the  interest  at  6%  pr.  an.  on  the  following: 

74.  * $36.57  for  3  mos  20  ds.     78.  $1673      for  8  mos  8  ds. 

75.  $2977  "  6    "     16  "      79.  $936       "  4  "  19  " 

76.  $9856  "  4    "     15  "      80.  $281       "  3  "  27  " 

77.  $2836  "  9    "    27  "       81.  $106.27 "  8  "  16  lt 
Answers:  $460.06  and  $103.40. 

*When  the  principal  is  small  and  the  time  long,  interest  may  be 
computed  on  cents. 


148          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Find  the  amounts  of  the  following,  and  be  particular  to 
add  the  cents  of  the  principal : 

Mos.  Days.  Mos.  Days. 

82.  $250.15  for  6   2       86.  $501.19  for  6  6 

83.  $380.67  "  10  10       87.  $219.12^  "  4  27 

84.  $900.90   "    19       3                88.     $369. 16f   «      8  12 

85.  $216.67   "     8     20                89.  $1220.00     «    10  10 
Answers:  $1871.11  and  $2408.91. 

161.    To  find  the  interest  for  years  at  6%  per  annum. 

Business  usage  allows  only  360  days  to  the  year,  which 
is  6  sixty  days;  hence,  the  interest  for  60  days,  multiplied 
by  6  times  the  number  of  years,  gives  the  result. 

90.  Find  the  interest  on  $120  for  1  year,  4  months  and 
20  days,  at  6%  per  annum. 

1.20  EXPLANATION. — The  interest  for  60  days  is  120  cents; 

8  for  1  year  and  4  months  it  is  8  times  120  or  960  cents; 

q  P~  and  for  20  days  it  is  J  of  120,  or  40  cents,  making  the 

'  sum  $10.00 — the  interest  required. 

$10.00  Ans. 

91.  Find  the  interest  of  $240  for  3  years,  4  months  and 
10  days.  Ans.  $48.40 

92.  What   is   the   interest   of  $1467.45    for   2   years,  6 
months  and  17  days?  Ans.  $224.21. 

Find  the  interest  of  the  following: 

93.  $321.00  for  2  years  3  months  15  days.* 

94.  $1767.00  for  7  years  4  mouths  21  days. 

95.  $897.25  for  3  years  6  months  27  days. 

96.  $898.57  for  2  years  7  months  25  days.f 

97.  $716.27  for  2  years  1  month       9  days. 
Answers:  $90.57,  $44.14,  $783.66,  $192.41,  $143.09. 

*Find  the  interest  for  2  years  4  months,  and  deduct  the  interest 
for  15  days. 

tCall  this  2  years  8  months,  and  deduct  the  interest  for  5  days. 


\ 


SIMPLE  INTEREST.  149 

Find  the  interest  on  the  following  : 

98.  $810.98  for  1  year    6  months     7  days. 

99.  $50.00  for  9  years  7  months  18  days. 

100.  §8.00  for  9  years  3  months  27  days. 
Answers:  $90.58,  $73.94,  $10.90,  $4.48. 

101.  $3140.79  for  1  year      7  months     7  days=? 

102.  $795.17  for  2  years     1  month      1  day  =? 

103.  $3.90  for  3  years     5  months  15  days=? 

104.  $1057.57  for  1  year    11  months  11  days^? 
Total,  $526.01. 

105.  $2674.57  for  1  year    8  months  21  days=? 

106.  $7143.45  for  2  years  1  month     18  days=? 

107.  $1742.67  for  1  year    9  months  13  days=? 

108.  $2100.00  for  2  years  1  month       1  day  ==? 

109.  $4109.85  for  1  year    6  months  17  days=? 
Total,  $2022.35. 

110.  $7856.00  for  1  year      1  month    29  days—? 

111.  $677.19  for  3  years     3  months     3  days=? 

112.  $287.17  for  1  year      7  months  16  days=? 

113.  $97.19  for  5  years  10  months  14  days=? 

114.  $10.10  for  1  year      3  months  19  d»ys=? 
Total,  $743.95. 

115.  $57.87  for  2  years  6  months  14  days=? 

116.  $120.14  for  7  years  7  months     7  days=? 

117.  $340.00  for  9  years  1  month    24  days=? 

118.  $1657.00  for  1  year    3  months  24  days=? 

119.  $769.75  for  2  years  3  months  18  days=? 
Total,  $487.40. 

. : . 

The  Teacher,  when  reviewing  these  exercises,  may  require  his 
class  to  compute  interest  on  cents,  as  is  usually  done  in  courts  of 
justice. 


150          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

162.  Having  the  interest  at  6%  per  annum  to  find  it  at 
any  other  rate,  all-quo ts  of  6  may  be  used. 

At    1%  it  will  be  £   of  that  at  6%. 

At    5%  it  will  be   -J-   less  than  at  6%. 

At    7%  it  will  be   -|-  more  than  at  6%. 

At    2%  it  will  be  £   of  that  at  6%. 

At    4%  it  will  be  1   less  than  that  at  6%. 

At    8%  it  will  be   J   more  than  that  at  6%. 

At    9%  it  will  be   £   more  than  that  at  6%. 

At  10%  it  will  be  ^  of  that  at  6%. 

120.  Find  the  interest  on  $250  for  1  year,  3  months 
and  20  days  at  7%  pr.  annum. 

$2.50  —Intercut  for  00  days  at  6%. 
?  I, = Number  of  GO  days. 

125  " 
1750 

20  <Zoya. 


a«  6%, 
3.263=/ttteresl  at  1%. 

$22.8±S=Interest  at  7%. 

or  $22.85. 

121.  §708.18  for  6  yrs  1  mo    6  ds  ©  9%  pr.  an.=? 

122.  $1000.00  for  4  yrs  2  mos  4  ds  @  7^  pr.  au.r=? 
Answers:  $438.10,  $292.44. 

123.  $340  for  2  yrs  2  mos  20  ds  @  2J%  pr.  an.=? 

124.  $000  for  3  yrs  4  mos  15  ds  @  6%%  pr.  an.=  ? 

125.  $850  for  1  yr    2  mos  12  ds  at  §\%  pr.  an.=? 
Total,  $237.22. 

Find  the  interest  of 

126.  $617.18  for  3  mos  18  ds  @  15%  pr.  an. 

127.  $460.74  for  2  mos    5  ds  @  18%  pr.  an. 

128.  $765.12  for  8  mos  16  ds  @  20%  pr.  an. 
Total,  $151.55. 


SIMPLE  INTEREST.  151 

Find  the  interest  on  the  following  at  10%  per  annum: 

129.  $710  for     92  days,        133.  $496  for     91  days. 

130.  $19^8  for     27  days.        134.  $671  for     80  days. 

131.  $8889  for  128  days.        135.  $100  for  104  days. 

132.  $75  for  117  days.        136.  $269  for     73  days. 

Total,  $351.47.  Total,  $36.91. 

163.  It  is  customary  for  bankers  to  lend  money,  and 
discount  by  the  month  instead  of  the  year.  This  percent- 
age is  easily  converted  into  6%  interest,  and  the  work 
performed  with  as  much  ease  as  before. 

1  %  per  month  is  12%  per  year,  or  2  times  6%. 
l-J-%  per  month  is  18%  per  year,  or  SHimes  6%. 

2  %  per  month  is  24%  per  year,  or  4  times  6%. 

Find  the  interest  on  the  following: 

137.  $65  for  80  days  @  2   %  per  month. 

138.  840  for  33  days  @  \\%  per  month. 

139.  $190  for  63  days  @  2   %  per  month. 

140.  $700  for  93  days  @  3   %  per  month. 
Total,  $77.20. 

Find  the  amount  of  the  following 

141.  $710  for  36  days  @  l-J-%  per  month. 

142.  $216  for  45  days  @  2   %  per  month. 

143.  $1800  for  57  days  @  lJ-%  per  month. 

144.  $560  for  14  days  @  6J%  per  month. 
Total,  $3367.89. 

When  computing  interest,  the  ingenious  student  will 
contrive  many  ways  for  abbreviating  his  work.  Some- 
times he  will  take  advantage  of  the  aliquots  of  100;  at 
other  times  he  will  transpose  the  terms,  and  consider  the 
days  as  dollars  and  the  dollars  as  days,  or  he  will  reduce 
the  rate  mentally  to  6%,  if.it  is  some  other  rate,  and  thus 
simplify  as  well  as  abbreviate.  For  instance,  in  the  ques- 


152          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

tion  138,  lie  might  consider  the  $40,  §120,  and  transposing 
the  terms,  which  could  be  done  instantly,  he  would  simply 
have  to  multiply  the  33  by  2,  making  66c,  the  answer. 

It  will  materially  abridge  the  operation  and  expedite  the 
labor,  if  the  learner  will  observe  to  avoid  the  use  of  all 
lines,  figures  or  marks  that  arc  not  absolutely  necessary. 
As,  for  instance,  when  using  aliquot  parts,  to  write  only 
the  results  of  division,  as  shown  in  the  following  example: 

145.  Interest  of  $321  for  2  years,  1  month  3.21 

and  22  days  at  10%  per  annum.  40  125 

EXPLANATION. — Mentally  it  is  found  that  there  are  12  J         1.07 
60  days  in  2  years  and  1  month,  to  multiply  by  which  we  107 

divide  by  8.     The  division  by  6  and  the  multiplication  ji  0^9 

by   10  were  performed   simultaneously,    giving   $08.836  — 

or  $08.84  as  the  answer.  68.836 

In  Bank  Discount,  the  learner  will  find  numerous  ques- 
tions upon  which  to  exercise  his  ingenuity,  arid  the  judi- 
cious teacher  will  encourage  him. 

The  method  of  finding  True  Discount,  and  other  more  dif- 
ficult calculations,  will  be  found  at  the  end  of  Banking. 

PARTIAL    PAYMENTS,    OR    PAYMENTS    BY 
INSTALLMENTS. 

164.  Notes,  bonds,  etc.,  drawing  interest,  are  sometimes 
paid  by  installments,  and  the  amounts  thus  paid  indorsed 
on  them.  The  legal  rule  for  computing  interest  on  install- 
ments may  be  expressed  thus: 

Apply  the  payment  to  the  discharge  of  the  interest,  and 
if  there  is  a  remainder,  subtract  it  from  the  debt.  When 
the  payment  is  less  than  the  interest  due,  it  is  not  applied 
to  the  discharge  of  the  interest  or  debt,  but  is  indorsed 
on  the  note  until  the  installments  exceed  the  interest. 
The  sum  of  the  installments  is  then  taken  from  the  amount 
due,  and  interest  computed  on  the  remainder  as  before. 


SIMPLE  INTEREST.  153 

1.  $576.  CINCINNATI,  Oct.  9,  1857. 

On  demand,  I  promise  to  pay  Robert  Ingles,  or 
order,  Five  hundred  and  seventy-six  dollars,  with  interest. 
Value  received.  SAMUEL  DUNNING. 

On  the  note  are  the  following  indorsements : 
Rec:d,  Dec.  16,  1857,  $100. 

"       Pel.  28,  1858,         3. 

"       July  27,  1858,    150. 
Required  the  amount  due  September  3,  1858. 

Yrs.        Mos.      J)s. 

From    1857     12     16 
Take    1857     10       9 
Difference,  2       7  or  67  days. 

$576.00— ^4  wowntf  of  note. 

6A3=Intercst  on  $576  for  67  days. 

$582.43=  Total  amount  due. 

to  be  subtracted. 


2.43= Balance  due. 


The  second  payment  is  less  than  the  interest  due  and 
no  calculation  is  required. 

From  December  16,  1857,  to  July  27,  1858,  is  7  months 
11  days. 

$482A3=:Balance  due. 

17  .!§—  Interest  for  7  mos.  11  days. 


$500.18—  .^moimtf  due. 
153.QQ=  Amount  of  payments, 

$347  '.18=  Balance  due. 
From  July  27  to  September  3,  is  38  days. 


2.W=Literest  for  38  days. 

due  September  3,  1858. 


154          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

2.  $650.  BOSTON,  June  3,  1848." 

For  value  received,  I  promise  to  pay,  on  demand, 
to  H.  Crooks,  or  order,  Six  hundred  and  fifty  dollars, 
with  interest  at  6%  per  annum.  J.  F.  DAVIS. 

Indorsements. 

Jan.  6,    1850,    $95. 
Oct.  13,  1850,    350. 
June   3,  1855,      12. 
Sept.  7,  1857,  paid  the  balance;  how  much  was  it? 

Aris.  $405.92. 

3.  On  a  note  drawn  September  23,  1857,  for  $650,  with 
legal  interest,  there  are  the  following  indorsements : 

Oct.      4,  $100. 
Nov.    3,         2. 
Dec.  11),  •    210. 
April  3,  1858,  the  balance;  how  much  was  it? 

Am.  $354,32. 

4.  On  a. note   drawn  October  3,  1856,  for  $1000,  with 
interest,  are  the  following  indorsements: 

Nov.  30,  $50. 

Jan.    6,  1857,  100. 
Feb.     9,      "  5. 

June    6,      "       190. 
Feb.  3,  1859,  the  balance;  what  was  the  amount? 


XVIII.  BANKING. 

164.  Banking  is  the  business  of  dealing  in  money. 
Banking  houses  borrow  and  lend  money,  receive  money 
for  safe  keeping,  and  exchange  the  money  of  one  country 
for  that  of  another.  There  are  National  Banks,  Public 
and  Private  Banks,  Banks  of  Deposit,  Banks  of  Issue  or 
Circulation  and  Banks  of  Discount. 


BANKING.  155 

A  National  Bank  is  one  which  issues  notes  secured  by 
Donds  of  the  United  States,  deposited  with  the  United 
States  Treasurer,  and  is  doing  business  under  the  authority 
of  the  General  Government. 

A  Public  Bank  is  one  that  is  owned  by  a  joint  stock 
company,  who  commit  its  management  to  some  of  their 
number  chosen  for  that  purpose.  These  persons  are  called 
President  and  Directors. 

A  Private  Bank  is  one  which  is  owned  by  one  or  more 
individuals,  who  attend  to  its  business  personally. 

Banks  of  Deposit  receive  the  ready  money  of  merchants 
and  others  for  safe  keeping.  Banks  of  deposit  also  loan 
money  on  interest  and  some  pay  interest  on  deposits. 

Banks  of  Issue  or  Circulation  manufacture  and  issue 
paper  money,  called  bank-notes  or  bills.  Many  of  these 
batiks  also  receive  money  on  deposit  and  do  a  discount 
business. 

Banks  of  Discount  lend  money  on  interest,  when  suita- 
ble security  is  given. 

The  security  required  by  banking  institutions  when 
loaning  money,  is  a  note  or  notes  from  the  borrower,  with 
the  names  of  one  or  two  responsible  persons  written  on 
the  back,  in  such  a  manner  as  to  bind  them  for  payment, 
should  the  drawer  of  the  note  fail  to  pay  it  at  the  proper 
time.  The  persons  who  sign  their  names  thus  are  called 
indorsers,  the  writing  the  indorsement,  and  the  person  who 
writes  the  note  and  signs  his  name,  the  maker  or  drawer. 

The  notes    given   by  borrowers  may  be  their   own*  or 

*  Notes  are  always  known  by  the  names  of  the  makers.  We 
Bpeak  of  A  B1s  note,  though  it  is  in  our  possession. 

To  hold  an  indorser,  he  must  be  notified  of  the  maturity  of  the 
note  at  furthest  on  the  next  business  day,  if  he  resides  in  the  same 
town  or  city ;  otherwise,  a  notice  should  be  mailed  to  him  within 
the  same  time. 


156          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

those  of  other  persons.  If  those  of  others,  the  borrower's 
indorsement  alone  is  all  that  is  usually  asked. 

The  person  to  whom  the  note  is  made  payable  is  called 
the  payee,  and  the  person  possessing  it,  the  holder.  By 
the  face  of  the  note  is  meant  the  side  containing  the  prom- 
ise; also  the  amount  for  which  it  is  drawn. 

A  negotiable  note  is  one  that  can  be  transferred.  To 
be  negotiable,  it  should  be  drawn  "to  the  order'1  of  the 
payee,  or  to  him  "  or  order"  or  "to  bearer."  In  the  first 
two  cases,  to  make  it  negotiable,  the  payee  would  have  to 
write  his  name  on  the  back.  To  the  third  class  belong 
bank-notes,  whic^  may  be  passed  by  any  holder  without 
indorsement. 

The  phrase  "value  received"  is  not  considered  essen- 
tial to  a  note,  though  it  is  well  to  insert  it. 

Days  of  Grace  are  three  days  allowed  to  the  maker  of 
a  note  beyond  the  stated  time  of  payment.  A  note  that 
is  drawn  payable  90  days  after  date,  is  not  legally  due 
until  the  93d  day. 

165.  MATURITY  OF  NOTES. 

Notes  falling  due  on  Sunday  or  a  national  holiday  are 
payable  the  day  before. 

A  note  dated  the  28th,  29th,  30th  or  31st  of  Jan- 
uary, at  one  month,  falls  due  on  the  last  day  of  February 
without,  or  on  the  3d  day  of  March  with  grace ;  and  a 
note  dated  the  last  day  of  February,  at  one  month,  falls 
due  on  the  28th  of  March,  if  not  a  leap-year ;  otherwise, 
on  the  29th  of  March,  plus  the  days  of  grace.  Hence,  a 
note  drawn  by  the  month,  and  dated  on  the  last  day  of  the 
month,  falls  due  on  the  same  day  of  the  month,  if  the 
latter  month  have  a  corresponding  day;  otherwise,  it  falls 
due  on  the  last  day  of  the  month.  Thus,  a  note  dated 


BANKING.  157 

November  30th,  at  two  months,  without  grace,  would  fall 
due  on  the  30th  of  January ;  at  three  months,  oil  the  28th 
or  last  day  of  February. 

166.  REMARKS  ON  NOTES. 

1.  A  note  need  not  be  dated  at  the  place  where  it  is 
drawn,  and  can  be  made  payable  at  any  particular  place 
the  parties  may  agree  upon;   but  to  hold  an  indorser,  de- 
mand must  be  made  at  the  particular  place  specified.     A 
note  is  also  good  if  dated  on  Sunday. 

2.  When  giving  a  note,  the  maker  ought  to  fix  the  place 
Df  payment — say  his  bank  or  place  of  deposit — if  the  payee 
be  agreeable. 

3.  In  New  York  and  other  States,  notes  draw  seven  per 
cent,  interest.     Notes  made  payable  in  those  States,  accord- 
ingly, draw  that  rate  of  interest  after  maturity. 

4.  If  made  by  more  than  one  person,  it  is  called  a  joint 
note,  or  joint  and  several.     It  is  a  joint  note  only,  unless 
words    are   used   to   indicate    individual    responsibility,  as 
"We  jointly  and  severally,"  etc. 

5.  It  is  customary  for  merchants  to  deposit  their  notes 
in  bank  for  collection,  the  bank,  by  this  means,  undertak- 
ing to  see  them  paid,  or  using  means  to  hold  the  indorsers. 

6.  A  note  drawn  payable  to  the  maker,  and  by  him  in- 
dorsed in  blank,  can  be  transferred — negotiated — without 
liability  to  the  holder. 

7.  When    taking    up    a    note — paying    it— the    drawer 
ought  to  require  the  indorsement  of  the  holder,  deface  his 
own  signature  and  file  the  note  away,  as  it  may  subserve 
the  purpose  of  a  receipt  at  some  future  time. 

8.  A  note  drawn  under  the  seal  of  the  maker  is  called 
a  bond. 

Indorsements  are    of  various    kinds,  and,  like   the   note 
itself,  they  require  no  stated  form  of  words. 


158          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

There  are  Blank,  Full,  Special  and  Restrictive  indorse- 
ments. 

A  Blank  Indorsement  is  made  by  the  payee  or  holder 
writing  his  name  on  the  back. 

A  Full  Indorsement,  or,  as  it  is  called,  an  indorsement 
in  full,  is  made  by  writing  such  a  transfer  as  "Pay  to  the 
order  of  A,"  or  words  to  that  effect,  and  signing  the  name. 

A  Special  Indorsement  is  one  made  to  suit  a  particular 
case,  as  when  the  indorser  wishes  to  free  himself  of  re- 
sponsibility should  the  maker  fail  to  pay  the  note.  "Pay 
the  the  contents  to  B,  or  order,  without  recourse  on  me." 
"Payee." 

A  Restrictive  Indorsement  restrains  the  pa}rment  of  the 
note  to  the  party  to  whom  it  is  indorsed,  as  "Pay  the  con- 
tents to  C  only."  This  kind  of  indorsement  does  not  limit 
the  payment  when  made  by  another  person  than  the  payee 
of  the  note. 

The  payee  is  called  the  first  indorser,  if  he  transfers  the 
note,  and  the  party  to  whom  he  transfers  it,  the  indorsee. 
Should  the  latter  again  transfer  it,  he  would  b«e  called  the 
second  indorser  and  the  party  to  whom  it  was  transferred, 
the  second  indorsee. 

Indorsements  ought  to  be  written  across  the  back,  with 
the  left  end  up,  as  shown  on  page  160. 

167.  PtEMARKS  ON  INDORSING. — 1.  A  holder  or  indorsee 
can  write  over  a  blank  signature  a  full  or  restrictive  in- 
dorsement. 

2.  A  holder   may  cancel   all   indorsements    (signatures) 
but  the  first,  and   may  cancel  a  full   indorsement,  except 
the  signature  of  the  first. 

3.  The   object  of  a  full  indorsement   is   to   prevent  its 
transfer  without  the  signature  of  the  holder. 

4.  A  note  may  be  transferred  after   it  is  due,  but  the 
indorser  is  not  then  liable. 


BANKING.  159 

5.  The  payee  or  his  agent  must  make  the  first  transfer. 

6.  Any  partner  of  a  firm  may  indorse   for   all.     After 
maturity,  it  ought  to  be  done  severally. 

7.  Agents  may  indorse  for  principals  thus: 

A  B,  Principal, 

By  C  D,  Agent. 

8.  An  indorsement  cancelled  by  mistake  does  not  dis- 
charge the  indorscr. 

FORMS  OF  PROMISSORY  NOTES. 

The  following  are  given  as  examples  for  the  learner  to 
to  copy.  The  printed  blanks  differ  from  these,  and  are 
easily  filled. 

A  NEGOTIABLE   NOTE. 

CLEVELAND,  Sept.  30,  1866. 

Thirty  days  after  date,  I  promise  to  pay  to  the  order 
of  J.  C.  Hutsiupiller  Five  hundred  sixty  T3^j-  dollars,  at 
the  First  National  Bank,  this  city,  value  received. 

$560^.  E.  R.  FELTON. 

A  NON-NEGOTIABLE  NOTE. 

MILWAUKEE,  Apr.  4,  1867. 

Ninety  days  after  date,  I  promise  to  pay  to  Gr.  W.  Nel- 
son One  thousand  dollars,  value  received. 
$1000.  R.  C.  SPENCER. 

A  JOINT  AND  SEVERAL  NOTE  WITH  INTEREST. 

CINCINNATI,  Aug.  9,  1867. 

On  demand,  six  months  after  date,  we  severally  and 
jointly  promise  to  pay  to  the  order  of  William  Otte  Six 
thousand  dollars,  value  received,  with  interest  at  six  per 
cent,  per  annum.  GEORGE  F.  SANDS, 

W.  J.  BREED. 


160          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
FORM  OF  INDORSEMENT. 


When  writing  the  following  notes,  the  learner  ought  to 
use  cap  paper,  so  that  the  notes  be  long  enough  for 
filing,  and  leave  a  margin  above  and  below  of  not  less 
than  one  and  a  half  lines. 


HOME  EXERCISES  IN  DRAWING  NOTES. 

Draw  notes  from  the  following  data: 

1.  Date,  Aug.  18,  1868. 

Place,  Louisville. 

Maker,  Yourself. 

Time,  On  demand. 

Face,  $678.14. 

Payee,  S.  E.  Peyton. 

3.  Date,       April  9,  1870. 
Place,      Pittsburgh.      . 
Drawer,  S.  S.  Sargent. 
Payee,     Yourself. 
Face,       $168.75. 
Time,      3  months. 


2. 

Date, 

Nov.  6,  1869. 

Place, 

Cincinnati. 

Drawer, 

Yourself. 

Payee, 

James  Moore. 

Fnce, 

$375.50. 

Time, 

90  days. 

4. 

Date, 

Dec.  3,  1869. 

Place, 

Louisville. 

Time, 

On  demand. 

Drawer, 

E.  P.  Bowers, 

Payee, 

Indorser, 

R.  Johnson. 

Face, 

$6745.15. 

BANKING. 


161 


5.  Date,         Oct.  19,  1860.  6.  Date, 

Place,        Wheeling,  Va. 
Drawer,    Archibald  Warren. 
Payee,       Yourself. 

Indorser,  

Face,         $1364. 
Time,         6  months. 
Indorsee,  Saml.  Adams. 


Nov.  29,  1867. 


Date, 

Place, 

Time, 

Face, 

Drawer, 

Payee, 

1st  Indorser,  , 

Indorsee,         Yourself. 


April  3,  1869. 
New  York. 
4  mouths. 
$1408.75. 
James  Todd. 
Robt.  Emmet. 


9.  Place, 
Date, 
Time, 
Maker, 

Payee,  

1st  Indorser,  James  Moore. 
2d  Indorser,  Yourself. 
Face, 


Mobile.  10. 

April  3,  1867. 
One  day  afterdate. 
S.  0.  Miner. 


11. 


Place,  Boston,  Mass. 

Date,  July  10,  1870. 

Time,  30  days. 

1st  Indorser,  Robt.  Penn, 
1st  Indorsee,  James  Moon. 


Place,  New  Orleans. 

Drawer,          E.  M.  Small. 
Payee,  H,  C.  Piner. 

1st  Indorser,  

1st  Indorsee,  Edward  Epply. 

2d  Indorser,  

Face,  $138. 

Date,         Jan.  1,  1867. 
Place,        Chicago. 
Time,         60  days. 
Drawer,     R.  E.  Baird. 
Payee,       J.  E.  Piner. 

Indorser,   

Indorsee,  J.  J.  Wood. 
Face,         $1637.00. 

Place,        San  Francisco. 
Date,         

Time,         4  months. 
Maker,      Henry  Adams. 
Indorsee,  Thomas  Orr. 
Payee,       Henry  Adams. 

Indorser,  

Face,         $450. 

Maker,  S.  Poor. 

Payee,  

2d  Indorser,    ..* 

2d  Indorsee,  Wm.  A.  Miller. 
Face,  $396.57. 


Draw  the  notes  necessary  for  the  following  transac- 
tions : 

12.  Take  H.  P.  Spike's  obligation,  at  1  day  after  date, 
for  price  of  a  house  worth  $1695.* 


*To  make  this  a  bond,  insert  the  word  "seal"  after  the  name 
14 


162  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

13.  Pay  S.   0.  Mooney  §167.87  with   your  note  at  30 
days,  bearing  interest. 

14.  Lend  Henry  Shotwell  §1687,  on  his   note,  in  such 
a  way  as  to  enable  you  to  collect  the  money  at  any  time. 

15.  Jno.    Emerson's    note,    our    favor,    indorsed    to    G. 
Young,  for  §675  at  90  days,  was  protested  for  non-pay- 
ment on  January  3,  1866.     What  was  its  form? 

16.  Renew  your  note  of  $190,  and  take  3  months'  time, 
allowing  interest  for  the  time  the  new  note  has  to  run. 

17.  Pay  Juo.  Sallust  §367,  on  acct.,  with  a  note  at  90 
days. 

18.  Indorse  C  D's  note  of  Sept,  3,  I860,  for  6600  at  90 
days   (your  favor),  in  such   a   way  that    you  will    not   be 
held  responsible  for  payment. 

19.  Give    W.   A.    Moore    your   note    at    3    months,  and 
draw  it  in  such  a  way  as  will  prevent  him  from  negotiat- 
ing it. 

20.  Exchange  notes  with  your  neighbor,  each  drawing 
at  90  days,  and  indorse. 

21.  Draw   a  copy  of  your   note  for  §675,  favor  of  H. 
Corncob,  at  3  months,  upon  which  you  had  to  pay  $2.47 
interest  on  Sept  19,  at  6%. 

22.  Pay   Saml.  Saul   Pickwick   the  balance  of  S678.50, 
you  owe  him,  by  giving  note  in  favor  of  Sam.  Weller  at 
60  ds. 

23.  Take  the  note  of  J.  Morgan  Henry,  Madison,  Ind., 
for   $360,  at  3  months,  and   secure   yourself  against  the 
statute  which  allows  "  relief,"  etc. 

(After  the  words  "value  received,"  insert  "without  any  relief 
•whatever  from  valuation  or  appraisement  laws."  For  form  of 
note,  see  page  202.) 

24.  Indorse   the   receipt  of  §167.87,  amount  paid  you 
on  E.  C.  Johnston's  note,  dated  Aug.  9,  1866,  and  drawn 
for  §974.35  at  6  months,  your  favor. 


BANKING,  163 


CHECKS— DRAFTS— BILLS  OF  EXCHANGE. 

A  Check  is  a  written  order  on  a  bank  or  its  cashier  for 
the  payment  of  money  in  its  possession  belonging  to  the 
party  making  it.  A  Draft*  is  similar  to  a  check,  but  is 
written  more  formally  and  may  be  drawn  upon  any  other 
than  a  banker;  and  a  Bill  of  Exchange  is  a  check  or  draft 
used  for  transmitting  money  to  distant  places. 

A  check,  like  a  promissory  note,  may  be  drawn  to 
bearer,  to  order,  or  be  made  payable  to  a  particular  per- 
son. When  given  to  strangers,  or  for  large  sums  of  money, 
they  should  be  drawn  to  order,  so  that  those  receiving 
them  would  have  to  indorse  them  before  payment.  In 
this  way,  they  may  also  be  used  to  subserve  the  purposes 
of  receipts. 

Checks  may  be  antedated  or  post-dated.  In  the  former 
case,  they  are  payable  on  presentation,  if  there  are  funds 
of  the  drawer  to  meet  them;  in  the  latter,  they  will  not 
be  payable  till  the  date  arrives.  Should  a  bank  refuse  to 
pay  a  check,  the  party  to  be  sued  is  the  maker,  or  trans- 
ferrer,  if  it  has  been  transferred. 

If  a  bank  pay  a  forged  check,  the  bank,  and  not  the 
person  whose  signature  is  forged,  has  to  sustain  the  loss, 
though  the  forgery  may  be  so  well  executed  that  it  caa 
not  be  detected  by  ordinary  inspection. 

When  a  check  is  so  carelessly  drawn  that  an  alteration 
may  be  easily  made,  the  loss  arising  must  be  borne  by 
the  drawer.  For  instance,  the  amount  to  be  paid  should 
commence  at  the  extreme  left  of  the  line,  and  the  part 
left  unfilled  be  written  with  a  curved  line  or  other  mark 
to  prevent  additions. 

*For  forms  of  draft,  see  Exchange. 


164          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
FORM  OF  CHECK. 


OUDERS— DUE-BILLS. 

A  mercantile  Order  is  a  request  for  the  delivery  of 
goods  or  money.  It  differs  from  a  draft  in  being  more 
simple  in  its  form,  and  its  being  usually  drawn  for  the 
payment  of  goods  instead  of  money.  The  following  is  the 
common  form: 

CINCINNATI,  Jarfy  1,  1867. 
MESSRS.  JOHN  SHILLITO  &  Co. 

Will  please  let  Samuel  Steele,  or  bearer,  have  goods  to 
the  amount  of  One  hundred  dollars,  and  charge  to  my  ac- 
count. 

$100^.  GEORGE  KIDD. 

To  be  more  specific,  the  kind  of  goods  and  the  price, 
wholesale  or  retail,  might  be  inserted. 

A  Due-bill  is  simply  an  acknowledgement  of  a  debt. 
It  is  usually  drawn  for  a  small  sum  and  to  settle  an  ac- 
count. 

Due-bills  are  considered  to  draw  interest  from  date, 
though  it  is  seldom  exacted. 


BANKING.  165 

FORM  OF   A  DUE-BILL. 

INDIANAPOLIS,  Aug.  3,  1868. 

Due  Henry  Ellsworth,  or  order,  Fifteen  dollars.     Value 
received. 
$15_QLQL>  HENRY  SPENCER. 

RECEIPTS. 

It  may  not  be  considered  improper  here  to  introduce  a 
few  forms  of  receipts,  as  they  are  essentially  connected 
with  the  preceding  subjects. 

A  Receipt  should  specify  what  it  was  given  for,  whether 
money,  goods,  note,  etc.,  the  amount  for  which  it  was 
given  and  the  date,  and  the  amount  should  be  in  writing. 
Receipts  for  sums  over  twenty  dollars  should  have  a  stamp. 

Receipts  may  be  made  on  bills,  notes,  etc.,  or  given 
separately,  and  need  not  be  of  any  particular  form. 

RECEIPT  ON   ACCOUNT. 

CINCINNATI,  Jan.  1,  1868. 

Received  of  Mr.  John  Cummins,  One  hundred  and 
twenty-five  dollars  and  23  cents  on  account. 

-.  JAMES  MORGAN. 


RECEIPT  FOR  MONEY  ON   NOTE. 

PITTSBURG,  Apr.  3,  1867. 

Received  of  Alex.  Cowley,  One  thousand  dollars,  to  be 
credited  on   his   note,   my  favor,  dated  Jan.  3,   1867,  for 
six  thousand  dollars. 
$1000.  W.  ALLAN  MILLER.* 


*A  payment  on  a  note  should  be  receipted  (indorsed)  on  the  back 
of  the  same,  and  a  statement  made  that  a  receipt  was  given. 


166          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

RECEIPT   IN   FULL. 

Received,   Cincinnati,   June   3,  1SG8,  of  Timothy   Hay, 
Thirty -five  -,YV  dollars,  in  full  to  date. 
$35.25.  THOMAS  E.  YOUNGMAX. 

RECEIPT  FOR  RENT  ON  ACCOUNT. 

BOSTON,  Feb.  27,  1867. 

Received  of  Mr.  Henry  G.  Judkins  Sixty-eight  dollars, 
on  account  of  rent  for  No.  6  Long  St. 
$68.00.  EDW.  FABER. 

RECEIPT  FOR  A  NOTE. 

PHILADELPHIA,  Jariy  30,  1869. 

Received  of  Mr.  James  Thompson,  his  note,  my  favor, 
at  ninety  days,  for  Five  hundred  dollars  to  balance  acct. 
$500.  JNO.  F.  GREEN,  JR. 

RECEIPT  FOR  MONEY  IN  ADVANCE. 

COLUMBUS,  Apr.  3,  1869. 

Received  of  J.  Q.  A.  Miller,  Forty  dollars,  in  advance, 
for  live  pork,  to  he  delivered  to  him  on  or  before  October 
1  1869,  at  7  cents  a  pound. 
$40.  JAS.  HOLLANDER,  SEN. 

RECEIPT  FOR  MONEY  RECEIVED  ON  ACCOUNT  OF  ANOTHER. 

CINCINNATI,  Oct.  10,  1866. 

Received  from  H.  D.  Brown,  on  the  account  II.  W. 
Sou  they,  Sixty-nine  y8^  dollars,  in  full  of  acct  to  1st  inst. 
$69^.  Y.  W.  COOK,  JR. 

RECEIPT  FOR  RENT  IN  FULL. 

NEW  YORK,  Sept.  12,  1866. 

Received  from  R.  Quinn,  Twenty-five  dollars,  in  full  for 
rent  of  house,  No.  96  Chestnut  St.,  to  9th  iust. 
$25.  JAS.  M.  HALL. 


BANKING.  167 

RECEIPT   FOR  MONEY  RECEIVED   BY  A  CLERK. 

NEW  YORK,  Aug.  3,  1870. 

Received  of  II.  Simon  One  hundred  dollars  on  acct. 
.  JAMES  MOORE. 

Per  JNO.  WOOD,  Clk. 


HOME  EXERCISES  IN  DRAWING  RECEIPTS. 

1.  Receipt  to  John  Roberts  for  $100  on  account. 

2.  Give  H.  C.  Parker  a  receipt  for  $50  in  full   of  ac- 
count. 

3.  Draw  a  receipt  for  $250T5,?g-  in  favor  of  C.  C.  Martin, 
for   his    note  of  this   date  in   settlement  of  account,  and 
draw  a  copy  of  the  note, 

4.  Receipt   to   Mrs.   T.   H.  Henshaw   for  $365,  in   part 
for  rent  of  house,  1968  Vine  Street. 

5.  Give  your  teacher  a  receipted  bill  for  257  barrels  of 
flour  at  $12.67  per  barrel;   drayage,  $8.50. 

6.  Let  William  Eduiuridson  have  your  receipt  for  $67, 
on  his  note,  of  the  3d  of  last  month,  at  6  months,  your 
favor;  and  draw  a  copy  of  the  note,  showing  a  receipt  to 
be  made  on  it.     Face  of  note,  $936. 

7.  M*ke  out  to  your  teacher,  this  date,  the  bill  on  page 
111,  and  take  a  note  in  settlement. 

8.  Receipt  for  $150   cash   on   the   first  bill,  page  112, 
making  it  to  William  Nelson. 

9.  As  clerk  of  II.  J.  Estcourt,  make  out  the  first  bill 
on  page  113  to  your  teacher,  allowing  him  a  discount  of 
5%  for  cash. 

10.  Give  your  due-bill  and  an  order  on  your  grocer,  to 
teacher,  for  tuition  one  session  of  5  months,  amounting  to 
$85.     Make  the  order  for  a  balance  of  $15.75  in  grocer's 
hands,  and  draw  all  the  papers. 


168 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


TIME   TABLE 

FOR  COMPUTING   INTEREST  AND   AVERAGE. 

Number  of  days  from  Is/  of  Jan-nary  to  any  other  day  of  the  year.    In  leap-years,  add  I 
to  the  days  after  26th  of  February. 


a 

£ 

o 

i 

j.Tiinmry  .... 

I  jFobrujir; 

• 
*i 

»*» 

T3 

g 

S» 
X 

«-l 
S3 

3 

0> 

,Day  of  Mo. 

^ 

e^ 
5T 

I  August  

cc 

4 

3 
1 

I 

n 

1  1  November 

i  December. 

I! 

Day  of  Mo. 

i  

:  1  : 

1 

0 

:51 

59 

90:120 

151 

1 

181 

212j243|273 

304 

334 

1 

2 

1 

32 

60 

91 

121 

152 

2 

182 

213 

244i274 

305 

335 

2 

3 

2 

33 

61 

92 

122 

153 

3 

183 

214 

245 

275 

306 

336 

3 

4 

3 

34 

62 

93 

123 

154 

4 

184 

215 

246 

276 

307 

337 

4 

5 

4 

35 

63 

94 

124 

155 

5 

185 

216 

247 

277 

308 

338 

5 

6 

5 

36 

64 

95 

125 

156 

6 

186 

217 

248 

278 

309 

339 

6 

7 

C 

37 

65 

96 

126 

157 

7 

187 

218 

249 

279 

310 

340 

7 

8 

7 

38 

86 

97 

127 

158 

8 

188 

219 

250 

280 

311 

341 

8 

9 

8 

3! 

67 

98 

128 

159 

9 

189 

220 

251 

281 

312 

342 

9 

10 

9 

40 

68 

99 

129 

160 

10 

190 

221 

252 

282 

313 

343 

10 

11 

10 

41 

69 

100 

130 

161 

11 

191 

222 

253 

283 

314 

344 

11 

12 

11 

42 

70  101 

131 

162 

12 

192 

223 

254 

284 

315 

345 

12 

13 
14 

12 
IB 

43 
44 

71  102132 
721031133 

163 
164 

13 
14 

193 
194 

224 
225 

255 
256 

285 
286 

316 
317 

341) 
347 

13 
14 

15 

14 

15 

73 

104 

134 

165 

15 

195 

226 

257 

287 

318 

348 

15 

16 

15 

46 

74 

105 

135 

166 

16 

196 

227 

258 

288 

319 

349 

16 

17 

1(5 

47 

75 

106 

136 

167 

17 

197 

228 

259 

289 

320 

350 

17 

18 

17 

48 

76 

107 

137 

168 

18 

198 

229 

260 

290 

321 

351 

18 

19 

18 

49 

77 

108 

138 

169 

19 

199 

230 

261 

291 

322 

352 

19 

20 

19 

50  78 

109 

139 

170 

20 

200 

231 

262 

292 

323 

553 

20 

21 

20 

51 

79 

110 

140 

171 

21 

201 

232 

263 

293 

324 

354 

21 

22 

21 

52 

80 

111 

141 

172 

22 

202 

233 

264 

294 

325 

355 

22 

23 

22 

53 

81 

112  1421173 

23 

203 

234 

265 

295 

326 

356 

23 

24 

23 

54 

82 

113|143 

174 

24 

204 

235 

266 

296 

327 

557 

24 

25 

24 

55 

83 

114 

144 

175 

25 

205 

236 

267 

297 

328 

-558 

25 

26 

25 

56 

84 

115 

145 

176 

26 

206 

237 

2681298 

329 

359 

26 

27 

26 

f)7 

85 

116 

146 

177 

27 

207 

238 

269 

299 

330 

360 

27 

28 

27 

58 

86 

117 

147 

178 

28 

208 

2391270 

3001331 

561 

28 

29 

28 

87 

118 

148 

179 

29 

2091240 

271  301 

332 

362 

29 

30 

29 

88 

119 

149 

180 

30 

210 

241 

272 

302 

333 

363 

30 

31 

30 

89 

150 

31 

211 

242 

303 

364 

31 

BANKING. 


169 


TIME  TABLE 

FOR  COMPUTING  INTEREST  AND   AVERAGE. 

Number  of  days  from  1st  of  July  to  any  other  day  of  the  year.    In  leap-years,  add  1  to 
the  days  after  28//t  of  February. 


b 

p 

s. 

o 

£ 

fT 

August  , 

^September  .. 

I  ^ 
i  g. 

o 
^ 

1  November  ., 

!  December... 

Day  of  Mo... 

[January  

I 

i  February.... 

|March  

> 

V 

£ 

"  | 
<*$ 

«-| 

Bl 

0 

0 

'Day  of  Mo... 

i 

1 

0 

31 

62 

92 

123 

153 

1 

1841215  243 

274 

304 

335 

1 

2 

1 

32 

63 

93 

124 

154 

2 

185 

216 

244 

275 

305 

336 

2 

3 

2 

33 

64 

94 

125 

155 

3 

186 

217 

245 

276 

306 

337 

3 

4 

*j 

34 

65 

95 

126 

156 

4 

187 

218 

246 

277 

307 

338 

4 

5 

4 

35 

66 

96 

127 

157 

5 

188 

219 

247 

278 

308 

339 

5 

0 

5 

36 

67 

97 

128 

158 

6 

189 

220 

248 

279 

309 

340 

6 

7 

6 

37 

68 

98 

129 

159 

7 

190 

221 

249 

280 

310 

341 

7 

8 

7 

38 

69 

99 

130 

160 

8 

191 

222 

250 

281 

311 

342 

8 

9 

8 

39 

70 

100 

131 

161 

9 

192 

223 

251 

282 

312 

343 

9 

10 

9 

40 

71 

101 

132 

162 

10 

193 

224 

252 

283 

313 

344 

10 

11 

10 

41 

72 

102 

133 

163 

11 

194 

225 

253 

284 

314 

345 

11 

12 

11 

42 

73 

103 

134 

164 

12 

195 

226 

254 

285 

315 

346 

12 

13 

12 

43 

74 

104 

135 

165 

13 

196 

227 

255 

286 

316 

347 

13 

14 

13 

44 

75 

105 

136 

166 

14 

1971228 

256 

287 

317 

348 

14 

15 

14 

45 

76 

106 

137 

167 

15 

198 

229 

257 

288 

318 

349 

15 

16 

15 

46 

77 

107 

138 

168 

16 

199 

230 

258 

289 

319 

J50 

16 

17 

16 

fc7 

78 

108 

139 

169 

17 

200 

231 

259 

290 

320 

351 

17 

18 

17 

48 

79 

109 

140 

170 

18 

201 

232 

260 

291 

321 

352 

18 

19 

18 

49 

80 

110 

141 

171 

19 

202 

233 

261 

292 

322 

353 

19 

20 

19 

50 

81 

111 

142 

172 

20 

203 

234 

262 

293 

323 

i&4 

20 

21 

20 

5182J112 

143 

173 

21 

204 

235 

263 

294 

324 

355 

21 

22 

21 

52 

831113 

144 

174 

22 

205 

236 

264 

295 

325 

356 

22 

23 

22 

53 

84lll4 

145 

175 

23 

206 

237 

265 

296 

326 

}57 

23 

24 

23 

54 

85  115 

146 

176 

24 

207 

238 

266 

297 

327 

358 

24 

25 

24  55 

86116 

147 

177 

25 

208 

239 

267 

298 

328 

359 

25 

26 

25 

5(5 

871117 

148 

178 

26 

209 

240 

268 

299 

*29 

J60j 

26 

27 

26 

57 

88 

118 

149 

179 

27 

210 

241 

269 

300 

330 

'361 

27 

28 

27 

58 

89 

119 

150 

180 

28 

211 

242 

2701301 

331 

362 

28 

29 

28 

59 

90 

120 

151 

181 

29 

212 

271!302 

332 

'363 

29 

30 

29 

60 

91 

121 

152 

182 

30 

213 

272 

303 

333 

364 

30 

31 

30 

61 

122 

183 

31 

214 

273 

334 

j  SI 

15 


170  NELSON'S  COMMON-SCHOOL  ARITHMETIC 

USE   OF   THE   PRECEDING  TABLES. 

1.  From  ~[st  of  July  to  9//i  of  June,  how  many  days? 
Opposite  9,  in  the  last  column,  is  342,  the  answer. 

2.  From  2d  of  October  to  17th  November,  liow  many  days? 
Opposite    2,  in    the    middle  column,  is    93,  in  October 

column,  and  opposite  17  is  139,  in  the  November  column. 
The  difference  is  46. 

These  tables  are  specially  adapted  for  averaging  ac- 
counts, taking  1st  of  January  and  1st  of  July  as  dates 
from  which  to  reckon. 

DISCOUNTING  NOTES. 

168.  Discounting  notes  consists  in  buying  them  at  less 
than  their  nominal  value,  or  the  amount  for  which  they 
are  drawn.     The  difference  between  the  nominal  value  and 
the  price  paid  is  called  discount. 

169.  Bankers  prefer  lending  money  on  short  time,  and 
by  the  day,  instead  of  by  the  month.     Notes  are  usually 
drawn    for    30,    60    or    90    days;    and    interest   is    always 
charged  on  the  days  of  grace. 

170.  There  are  two  kinds  of  discount:    True  Discount, 
which  is  the  interest  paid  in  advance  on  the  present  value 
of  a  note,  and  Bank  Discount,  which  is  interest  paid  in 
advance   on   the  face  of  the    note.     The  latter   resembles 
compound  interest,  as  it  is  interest  on  both   interest  and 
principal.* 

When  a  note  is  discounted  in  bank,  the  interest  of  the 
note  for  the  time  it  has  to  run,  and  at  the  banker's  rates, 
is  deducted  from  the  sum  called  for  by  the  note.  The  bal- 

*The  present  worth  of  a  note  drawn  for  $100,  payable  in  a  year 
at  6  per  cent.,  is  $94.84,  and  the  interest  is  $5.56;  that  is,  the  prin- 
cipal and  interest  together  are  equal  to  $100,  or  the  face  of  the  note; 
BO  when  a  banker  discounts  from  the  face  of  a  note,  he  discounts  ou 
both  principal  and  interest. 


BANKING.  171 

ance  is  called  the  proceeds.  This  species  of  discount  is 
therefore  reckoned  in  the  same  way  as  interest.  Bankers 
reckon  interest  on  every  day  intervening  between  the  day 
of  discount  and  that  of  maturity,  including  the  latter. 

171.  When  a  note  is  drawn  by  days,  subtract  the  ex- 
pired term  from  the  number  of  days  for  which  it  is 
drawn,  plus  the  days  of  grace ;  but  when  drawn  by  the 
month,  first  find  the  day  of  maturity  and  reckon  the 
whole  number  of  days  from  the  day  of  discount  to  that 
date. 

1.  Discounted  on  the  day  of  date,  how  much  discount 
jhould  be  deducted  from  a  note  of  $500  at  90  days?* 

$5. 00= Interest  for  GO  days. 
2.50==       "        "    30     " 
.25^       «        «       3     "       (grace.) 

Arts.  $7.75 

2.  $1500.  COLUMBUS,  Jan.  8,  1859. 

Sixty  days  after  date,  I  promise  to  pay  Messrs. 
M'Ewen  and  Banfill  One  thousand  five  hundred  dollars, 
value  received.  GEO.  K.  TENNEY. 

Required,  the  discount  at  6%  per  an.  Ans.  $15.75. 

3.  $3500.  WHEELING,  Oct.  3,  1858. 

Ninety  days  after  date,  I  promise  to  pay  John 
M'Culloch,  or  order,  at  First  National  Bank,  three  thou- 
sand five  hundred  dollars,  value  received. 

MILO  G.  DODDS. 
Required  the  proceeds  at  6%  per  an.         Ans.  $3445.75 

4.  Find  the  proceeds  of  a  note  for  $120  at  GO  days  at 
\%  Per  month. 

5.  Required  the  proceeds  of  a  note  dated  Jan.  1,  18Q6J 
and  drawn  for  $575.75  at  90  days. 

*Wken  the  rate  is  riot  named,  six  per  cent,  per  annum  is  under- 
stood. 


172          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

6.  What  is  the  bank  discount  on  a  note  of  $450  for  60 
days  at  2%  per  month?* 

7  to  12.  Find  the  proceeds  of  the  following: 

On  $850  at  30  days  at  \\%  per  month;  on  $1678.25 
at  90  days  and  \\%  per  mouth;  on  $670  at  60  days  and 
2%  per  month;  on  $1749.57  at  90  days,  1J%  per  month; 
on  $688  at  90  days,  Z\%  per  month;  on  $6784  at  60 
days,  If  %  per  month. 

Answers:  $118.74,  $566.82,  $344.22,  $18.90,  $1668.19, 
$6534.69,  640.01,  $1613.22,  $6594.69,  $641.86. 

Find  the  discount  on  the  following: 

13.  $1310.00  for  60  days  @  2   %  per  month. 

14.  $746.87  for  90  days  @  \\%  per  month. 

15.  $219.56  for  30  days  @  1    %  per  month. 

16.  $1867.25  for  20  days  @  1\%  per  month. 

17.  $1367.00  for  15  days  @  3   %  per  month. 

Total,  $152.57. 

18.  A  note  drawn  on  February  13,  1866,  for  $900,  at 
90  days,  was  discounted  on  March  23,  at  2%  per  month; 
how  much  was  paid  to  the  borrower?  Am.  $867. 

19.  What  proceeds  should  be   paid  on  a  note  of  $346: 
at  90  days,  drawn  on  November  3,  and  discounted  on  De- 
cember 7,  at  \\%  per  month?  Ans.  $335.79.' 

20.  A  note  for  $689,  made  September  9,  payable  in  60 
days,   was   discounted   on   October   5,   at  2%    per   month; 
what  was  the  discount?  Ans.  $16.99.i 

While  the  cents  in  the  principal  are  rejected  in  comput- 
ing interest  and  discount,  they  are  always  reckoned  when 
finding  the  amount  or  proceeds. 

*Such  questions  as  these  may  be  abbreviated  by  mentally  in- 
creasing the  days  or  principal  in  the  ratio  that  the  rate  is  to  6  per 
cent.  In  this  case,  reckon  interest  on  $450X4  or  $1800  at  G  per 
cent,  per  annum. 


BANKING.  173 

Required  the  discount  on  the  following: 

Face  of  note.  Date.  Time.      When  disc'd.     Rate  of  discount. 

21.  $167.50  Jan.     3,  1869,  60  ds,  Feb.     7,  2   %  per  mo. 

22.  $9876.00  Feb.    7,  1869,  90  ds,  Mar.  12,  2J%  per  mo. 

23.  $789.00  Juu.  18,  1869,  30  ds,  July    3,  l\%  per  mo. 

24.  $1897.00  Feb.  21,  1869,  90  ds,  Apr.     1,  l|%  per  mo. 

Total,  $555.24. 
Find  the  proceeds  of  the  following : 

25.  $676.37  Apr.     3,  1869,  90  ds,  May  9,  2   %  per  mo. 

26.  $679.39  Mar.      9,  1869,  30  ds,  Apr.  3,  ty%  per  mo. 

27.  $7168.00  June  13,  1869,  60  ds,  July  9,  \\%  per  mo. 

28.  $816.37  Aug.  12,  1869,  30  ds,  Sep.    6,  Z\%  per  mo. 

Total,  $9172.40. 

29.  A   note  for  $4378.35,  dated   February   1,  1867,  at 
4  months,  was  discounted  on  May  16  at  10%  per  annum; 
[required  the  proceeds. 

SOLUTION. — This  note  falls  due  June  4.  From  May 
16  to  that  date  is  19  days,  giving  for  discount  on  $4378, 
823.11.  Proceeds,  $4355.24. 

30.  Required  the  proceeds  of  a  note  dated  September  3, 
1867,  for  $396.87,  at  3  mouths,  and  discounted  on  Octo- 
ber 5,  at  \\%  per  month. 

31.  A  note  dated  September  30,  at  3  months,  and  drawn 
or  $1367.56,  was    discounted   on  October  1,  at  2%   per 
month;  what  were  the  proceeds? 

32.  Required  the   proceeds  of  a  note   dated  December 
31,  1867,  drawn  for  $5363.75,  at  2  months,  and  discounted 
February  1,  at  \\%  Per  month? 

33.  A  note  for  $1000,  dated  Februray  28,  1866,  at  2 
months,  was  discounted  on  March  20,  at  1%  per  month; 
required  the  proceeds. 

Answers:  $986.00,  $384.56,  $1282.75,  $5296.70  $529.50, 


174          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
Find  the  proceeds  of  the  following: 

Fnre  of  note.  Date.  Time.        When  clisc'd.    Kate  of  discount. 

34.  $2676.00,  Jan.  9,  1869,  90  days,  Feb.  1,  \\%  pr  mo. 

35.  87187.00,  Feb.  3,  1869,  60  days,  Mar.  13,  l%%  pr  mo. 

36.  -$768.21,  Mar.  6,  1869,  30  days,  Apr.  3,  2  %  pr  mo. 

37.  $314.00,  Apr.  7,  1869,  90  days,  May  15,  2   %  pr  mo. 

Total  discount,  $181.95. 

Ain't  of  note.  Date.  Time.       When  discounted.        Rate  of  disc't. 

38.  $6785,  Dec.  6,  1868,  6  mo.  Dec.  29,     2  %  pr  mo. 

39.  $3748,  Jan.  3,  1868,  5  mo.  Feb.  3,  1868,  2±%  pr  mo. 

40.  $6983,  Mar.  9,  1868,  4  mo.  June  8,  1868,  l\%  pr  mo. 

Total  proceeds,  $16272.70. 

Amount.        "Date.  Time.     When  disc'd.     Rate  of  discount. 

41.  $3784,  May     6,  2  mo.  July     3,  2   %   per  mo. 

42.  $6987,  Jun.     8,  3  mo.  Aug.  27,  \\%   per  mo. 

43.  $7854,  July  24,  4  mo.  Sept.  17,  1    %   per  mo. 

Total  proceeds,  $18371.58. 


XIX.  TRUE  DISCOUNT. 

172.  TRUE  DISCOUNT  is  the  difference  between  the  pres- 
ent worth  of  a  note  and  the  amount  for  which  it  is  drawn. 

The  present  worth  of  a  note  or  bill  due  at  a  future  time 
without  interest,  is  such  a  sum  as  would,  if  put  at  inter- 
est for  the  same  time  and  rate,  amount  to  the  debt;  and 
the  difference  between  this  sum  and  the  debt  is  the  discount. 

Except  in  courts  of  justice,  this  kind  of  discount  is  sel- 
dom used,  business  men  preferring  bank  discount  for  its 
simplicity. 

DISCOUNT  ON  INTEREST-BEARING  NOTES. — Bankers  dis- 
count off  the  value  of  the  note,  including  interest,  at  matur- 
ity ;  while  Real  Estate  Brokers  discount  off  the  face  of  the 
note,  plus  the  accrued  interest,  at  the  time  of  discount. 


TRUE   DISCOUNT,  ETC.  175 

1.  What  is  the  true  discount  on  a  note  of  $700  for  90 
days  at  6%? 

The   amount   of  a   dollar   for   93   days   is   $1.0155,   by 
which,  if  we  divide  $700,  we  will  find  the  present  worth. 
OPERATION.     $1.0155)700.0000(689.315 
60930 

NOTE.— The  interest,  on  $1    for  90  clays  is  90700 

0155.     The  present  value  of  $1.0155,  for  93  81240 

days   is,    therefore,  $1,   and,  accordingly,   the  "     qjrn (\ 

present  value  of  $700  for  93  days  is  $700  di-  91395 
rided  by  $1.0155  or  $089.31,  and  the  discount 


$700—  $689.31,  or  $10.09.  32050 

30465 

PROOF.— The  interest  on  $689.315,  93  days,  15850 

s  $10.683,  which,  if  added  to  the  principal,  10155 

will  give  $699.999  or  $700. 


56950 
50775 
The  pupil  can  prove  his  calculations  by  interest. 

2.  What  is  the  true  discount  on  a  note  of  $575  for  90 
days  at  6%? 

3.  What  is  the  true  discount  on  a  note  of  $137.09  for 
90  days  at  6%? 

4.  What  will  be  the  proceeds  of  a  note  of  $1878.67  at 
90  days,  true  discount? 

5.  A  note  for  $485.44,  payable  in  30  days  after  date,  is 
worth  how  much,  true  discount? 

173.  To  find  the  FACE  OF  A  NOTE,  when  the  proceeds, 
time  and  rate  are  given. 

6.  Required  the  principal  wlien  the  proceeds  are  $275.23, 
le  time  93  days  and  the  rate  2%  per  month. 

Interest  on  $1  for  93  days  at  2%  per  months. 062. 

Proceeds  of  $1=$1.000— .062=.938. 

Since  there  are  as  many  dollars  in  the  principal  as  the 
proceeds  of  $1  is  contained  times  in  the  proceeds  given, 
S275.230-r-.938  will  give  the  principal  required,  $293.42+. 


176          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

PROOF.— Interest  on  $293.42  for  93  days  at  2%  per 
month=18.19-|-,  which,  subtracted  from  $293.42.,  leaves 
$275.23,  the  proceeds. 

7.  The  proceeds  are  $212.60,  time  63  days,  rate  1J% 
per  numth ;  required  the  principal.* 

8.  What  principal  will  realize  $120  proceeds  in  6  months 
at  10%  per  annum? 

9.  The  time  is  three  months,  rate  10%   per  an.,  pro- 
ceeds $168.97;  what  is  the  principal? 

10.  The  rate  is  12%  per  annum,  proceeds  $693.75,  time 
4  months;  required  the  principal. 

174.  To  find  the  RATE  PER  CENT.,  when  the  principal, 
interest  and  time  are  given. 

11.  The  principal  is  $300,  time  60  days,  interest  $5; 
required  the  rate. 

Interest  on  $300  for  60  days  at  6%=r$3.  At  1%=.50. 
It  is  obvious  that  the  rate  will  be  as  great  as  the  number 
of  times  1%  is  contained  in  the  interest  given.  Hence, 
$5.00-f-50=the  rate,  10%. 

PROOF.— Interest  on  $300  for  60  days  at  10%  =$5. 

12.  The  principal  is  $396.15,  time   13  months  9   days, 
interest  $26.34,3;  required  the  rate. 

13.  What  is  the  rate  per  cent,  on  $144  for  5  days,  when 
the  interest  is  24  cents? 

14.  Eequired  the  rate  on  $250  for  60  days,  when  the; 
interest  is  $3.50. 

175.  To  find  the  TIME,  when  the  principal,  rate  per  cent, 
and  interest  are  given. 

Grace  being  allowed  only  on  notes  and  drafts,  where 
neither  is  named,  it  is  not  reckoned. 

*The  learner  can  prove  liis  work  by  computing  interest  on  the 
principal  found. 


GENERAL  EXERCISES  177 

•K-t 

15.  The  principal  is  $1440,  rate  10%  per  annum,  inter- 
est $37. 50;  required  the  time. 

Interest  on  $1440  for  1  day  at  10%— 40  cents. 

Since  there  are  as  many  days  as  the  interest  for  1  is 
contained  times  in  the  interest  given,  $37.50-^40=93^  or 
94  days. 

PROOF. — Interest  on  $1440  for  94  days  at  10%  per 
annum=$37.60.* 

16.  The  principal  is  $1674,  rate  2%  per  month,  interest 
$59.87 ;  required  the  time. 

17.  In  what  time  will  a  note  for  $600,  at  6%   per  an- 
num, draw  $27.50  interest? 

1.8.  A  note  for  $375  drew  $21  interest  at  6%  per  an- 
num; how  long  did  it  require  to  do  it? 

19.  A  merchant  wishes  to  know  the  time  it  will  take 
a  balance  of  $917.50  to  make  $60.80,  with  interest  at  10%. 

GENERAL  EXERCISES. 

1.  What  is  the  bank  discount  on  a  note  of  $375,  drawn 
at  90  days,  at  \%  per  month? 

2.  What  amount  of  proceeds   should  I  receive  from  a 
note  of  $796,  drawn  at  60  days,  2%  per  month? 

3.  What  is  the  present  value  of  a  note  drawn  for  $600 
at  30  days? 

4.  What  amount  of  money  should  I  receive  on  a  note 
of  $675,  discounted  at  35  days  (having  35  days  to  run), 
\\%  per  month? 

The  Teacher  can  increase  these  exercises  to  any  extent,  as  the 
pupils  have  to  furnish  proof  of  their  work;  hence,  only  a  few  have 
been  given  under  each  artlale. 

*  Interest  is  never  reckoned  on  the  fraction  of  a  day,  hence  the 
difference. 


178          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

5.  June  3,  discounted  my  note  of  $350  at  10%,  having 
30  days  to  run ;  required  the  discount. 

6.  February  6,  1858,  had  A.  Seers's  note  of  $500,  dated 
20th  December,  1857,  discounted  at  \\%  per  month,  time 
to  run,  33  days;  what  were  the  proceeds? 

7.  June  3,   1858,  discounted   my  note  of  December  9, 
1857,  drawn  for  $1000,  at  12  months,  10%;  what  had  I 
to  pay? 

8.  October  3,  lifted  my  note  of  17th  February,  drawn 
for  $1000,  at  60  days;  what  amount  of  money  had  I  to  pay? 

9.  July  30,  had  M.  Norton's  note  of  $360,  dated  23d 
July,  at  90  days,  discounted  at  2%  per  month,  what  sum 
of  money  (Hd  I  receive? 

Answers:  $596.72,  $11.63,  $762.57,  $663.19,  $1027.50, 
$2.92,  $339.36,  $946.67,  $491.75. 


XX.  COMPOUND  INTEREST. 

176.  IN  Compound  Interest,  the  interest  is  converted 
into  principal,  every  quarter,  half  year  or  year.  Capital 
is  thus  more  rapidly  increased  than  by  simple  interest. 

Any  person  acquainted  with  the  principles  of  simple 
interest  will  readily  understand  how  to  compute  this. 

When  there  is  a  settlement  of  accounts  between  the 
parties,  after  interest  has  become  due  and  interest  is 
charged  in  the  settlement,  interest  may  be  allowed  upon 
the  balance  found  due  by  the  settlement.  So  an  agree- 
ment, after  interest  is  due,  to  turn  it  into  principal  is  valid. 
Where  there  is  a  contract  between  the  parties  for  the  pay- 
ment of  interest  annually,  if  not  paid,  simple  interest  may 
be  allowed  upon  the  interest  from  the  time  it  is  due.* 

*Swan. 


COMPOUND  INTEREST.  179 

1.  What   is    the    compound    interest    on    $1000   for  2^ 
years  at  6%,  payable  semi-annually  (half-yearly)? 

The  interest  of  §1000  for  6  mos.,  $30.00 

Add  the  principal,  1000.00 

Amount  for  6  mo*.,  1030.00 

Interest  on  §1030  for  6  mos.,  30.90 

Amount  for  1  .year,  $1060.90 

Interest  on  §1060.90  for  6  mos.,  31.827 

Amount  for  18  mos.,  $1092.727 

Interest  on  $1092.727  for  6  mos,  32.78181 

Amount  for  Z  years,  $1125.50881. 

Interest  on  $1125.50881  for  6  mos.,          33.76526' 

Amount  for  2  years  6  mos.,  $1159.27407 

By  deducting  the  prittopa/,  1000.00 

We  have  the  comp\l  int.  for  2J  yrs.,      $159.27 
t 

2.  What  is  the  compound  interest  and  amount  of  $672 

for  4  years,  at  6%  per  annum? 

By  computing  interest  on  $1  for  a  number  of  years,  it 
will  be  found  that  the  second  amount  is  equal  to  the 
square  of  the  first ;  the  third  amount  to  the  cube  of  the 
first;  the  fourth  amount  to  the  fourth  power  of  the  first; 
each  power  corresponding  to  the  number  of  years.  Hence, 
to  find  the  amount  of  any  principal  for  any  number  of  years, 
it  is  only  necessary  to  multiply  the  principal  by  the  amount 
of  $1  for  the  time  and  rate.  Taking  the  last  example, 
1.064=the  amount  for  4  years,  which,  multiplied  by  672= 
.  amount  required. 

3.  Find  the  amount  of  $375  for  20  years,  at  6%  com- 
pound interest,  reckoned  annually. 

At  6  per  cent.,  money  will  double  itself  in  11  years,  10  months 
and  21  days;  at  5  per  v,eut.,  in  14  years,  2  months  and  15  days;  at 
8  per  cent.,  in  23  yeai'6,  5  months  arid  lOJ  days. 


180          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

4.  Required  the  compound  interest  on  $600  for  5  years 
at  1%. 

Answers:  $241.53,  $827.68,  $848.38,  $176.38,  $137.50. 


XXI.  AVERAGE. 

177.  AVERAGE  signifies  mean  or  medium.     An  average 
number  or  quantity  is  one  which  has  an*  intermediate  value 
between  two  or  more  numbers  or  quantities.     The  average 
between  3,  4  and  5  is  4. 

178.  The  average  may  be  found  by  dividing  the  sum 
of  the  quantities  by  their  number. 

1.  Find  the  average  between  3,  4,  5  and  8. 

3  EXPLANATION. — In    these  four  numbers   there   are  20 

4  parts ;  in  each  of  4  number  there  are  as  many  equal  parts 

5  as  4  is  contained  in  20,  viz.,  6. 
_8 

4)20 
5 

2.  What  is  the  average  price  of  5  cows,  which  cost,  re- 
spectively, $50,  $60,  $70,  $80  and  $90? 

3.  The  wages   of  9   hands   in  a  factory  are  $5,  $6,  $7, 
$8,  $10,  $11,  $12,  $15  and  $16,  respectively;  what  is  the 
average  ? 

4.  Required  the  average  length  of  the  following  pieces 
of  calico:  37  yds.,  35  yds.,  31  yds.,  30  yds.,  32  yds.,  27 J 
yds.,  291  ^1^  3QJ  yds.,  29|  yds. 

5.  Traveling  5   days,  at   the   rate  of  18  miles  the  first 
day,  20  the  next,  20^  the  next,  22J  the  next  and  25  the 
next,  what  was  the  average  speed  per  day? 

Answers :  70,  3%  10,  21^,  21TV 


AVERAGE.  181 

EQUATION  OF  PAYMENTS. 

179.  When  average  is  applied  to  the  settlement  of  ac- 
counts, the  process  is  called  Equation  of  Payments  or 
Equation  of  Time. 

Merchants  and  manufacturers  sometimes  sell  their  goods 
on  credit,  the  time  varying  from  three  to  nine  months, 
and  their  customers  making  numerous  purchases  before 
settlement.  The  object  of  equation  of  payments  is  to  ob- 
tain an  average  date  of  payment  for  these  purchases. 

6.  I    owe    $3    payable    in    2    months,  $4   payable  in    3 
months  and  5  payable  in  4  months ,  what  will  be  the  av- 
erage time  of  payment  for  the  whole  amount? 

The  use  of  $3  for  2  mos^that  on  $1  for  3x2=  6  mos. 

The  use  of  $4  for  3'mos=that  on  $1  for  3x4=12  mos. 

The  use  of  $5  for  4  mos=that  on  $1  for  4x5=20  mos. 

$12  38  mos. 

Hence,  the  use  of  $12  for  those  several  months  is  equal 

to  that  of  $1  for  38  months;  for  12,  it  will  be  -^  as  much, 

or  3^  months  or  3  months  5  days. 

7.  A  merchant  sells  a  bill  of  goods  amounting  to  $4000, 
to  be  paid  as  follows:  $400  in  30  days,  $600  in  60  days, 
$1000  in  90  days  and  the  balance  in  4  months,  or  120 
days;  what  would  be  a  mean  or  average  time  of  payment 
for  the  whole? 

A  credit  of  $400  for  30  ds.  is  the  same  as  a  credit  on  $1  for  12000  ds. 
«  600   "    60  "      "          "         "       "  1    "    36000  " 

"          1000   "    90  "      "          "        "      "  1    "    90000  " 

"          2000   "  120  "      "          "        "      "  1   "  240000  " 

4000  378000 

On  $1  there  is  a  credit  for  378000  days. 
On  $4000,  there  is  a  credit  for  378000  days  divided  by 
~94|  days. 


182          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

That  is,  the  $4000  might  be  paid  in  94J  days,  or  on 
the  95th  day,  without  either  party  sustaining  loss  by  in- 
terest. 

8.  A  merchant  sells  goods  to  the  amount  of  $1700,  $500 
payable  in   60   days,  $300   payable  in  90   days  and   $900 
payable  in  30  days  ;  what  is  the  average  time  of  payment 
of  the  whole? 

9.  Sold  a  bill  of  goods  amounting  to  $700,  J-  of  which 
is  payable  in  90  days,  ^  in  4  months  and  J-  in  6  months; 
required  the  average  time  of  payment* 

180.    To  find  the  average  date  of  purchase. 

10.  Purchased  goods  as  follows;  what  was  the  average 
date  of  purchase? 

December  31,  a  bill  of  $300;  January  3,  a  bill  of  $100; 
January  9,  a  bill  of  $200,  January  18,  a  bill  of  $800; 
January  23,  a  bill  of  $500. 

300  X    0  —  EXPLANATION.  —  The  first  was  due  at  the 

100X    3—      300      time  of  purchase;    the   second,   three   days 
200  X    9  —   1800      after;  the  third,  nine  days  after,  etc. 
800  X  18=14400         REMARK.  —  If  the  amounts  above  were  equal, 
500X23=11500      and  the  intervals  also  equal,  the  average  date 
of  purchase  would  be  on  Jan.  9,  because  it  is 


1900  o 

'  __  midway  between  the  first  and  last  dates. 

*W 

Or  15  days  after  December  31,  the  date  of  the  first  pur- 
chase, which  brings  the  time  up  to  January  15. 

If  these  debts  had  been  contracted  on  a  credit  of  three 
months,  a  note  dated  January  15  would  be  given  to  settle 
the  bill. 

The  time  tables  (pp.  168,  169)  are  admirably  adapted  for 
averaging,  prepared  as  they  are  for  the  two  common  pe- 
riods of  settlement,  January  and  July.  For  account  sales, 
the  book-keeper  should  prepare  similar  ones  for  each  month. 

*For  answers,  see  end  of  chapter. 


AVERAGE.  183 

11.  The  following  goods  were  sold  on  a  credit  of  90 
days : 

Required  the  average  date  of  purchase,  or  date  of  note. 

Jan.     1,  Invoice  of  Coffee 81000.00 

Jan.     6,        "          "  Sugar 3500.00 

Mar,    9,        "         "  Sunds 9734.00 

Mar.  13,        "         "      "      976.50 

Apr.    3,       "         "      "      1037.00 

§16247.50 

Required    the   date   of  maturity   of  a   3   months'  note, 
grace  included. 

12.  Sept.     3,  Invoice  of  Calicoes $3150.00 

"      19,       «         "Muslins 1174.00 

"      20,       "         "   Silks 3500.00 

Oct.    19,        "         "  Sundries 1743.00 

89567.00 

Find  the  equated  time  of  payment  for  the  following,  or 
date  of  a  sixty-days'  note : 

13.  Apr.     3,     §167.25*  14.  May     7,     $674.40 

9,  374.00  Jun.     7,  168.37 

"  19,  176.00                      .      "     10,  370.20 

"  20,  371.00  «     15,  167.00 

"  25,  197.87  "     19,  679.60 

"  30,  300.00  July  23,  679.45 

May  9,  150.57  Aug.  18,  993.18 

"  23,  720.18  "     19,  875.57 

181.    Wlien  goods  are  purchased  at  different  dates  and 
on  different  lengths  of  credit. 

15.  Purchased  the  following  bills   of  merchandise;  re- 

*  When  the  cents  are  under  50,  reject  them;  otherwise,  add  a  dol- 
lar to  the  dollars. 


184          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

quired  the  average  date  of  maturity,  or  the  equated  time 
of  payment  for  all  : 

Apr.  3,  a  bill  of  $250  on  3  months'  credit. 

Apr.  9,        "     "     157    "    6       "  " 

May  7,        "     "     250    "    4       "  " 

Jun.  9,        «     "     320    "    2       "  " 

If  we  substitute  the  date  of  maturity  of  each  of  these 
bills  for  the  date  of  purchase,  and  arrange  them  in  the 
order  of  time,  we  shall  have  a  problem  in  all  respects  sim- 
ilar to  those  under  last  Art. 

The  first  bill  falls  due  July  3;*  the  second,  October  9; 
the  third,  September  7;  and  the  fourth,  August  9.  Ar- 
ranged in  the  order  of  time,  they  appear  thus  : 

July  3,  $250  X      2—      500  By  the  July  table  we  find  the 

Aug.  9,  §320  X    39—12480  difference  of  time  to  be  2,  39,  68 

Sep.    7,  $250  X    68—17000  and  100  from  the  first,  and  the  av- 

Oct.    9,  $157  X  10d—  15700  era£e  time>  46  an(i  over  a  half,  or 

47  **?*'•  47  da*s  after 


977  )45680(46  • 

3908  «  August  17. 

6600 
5862 

"738 

When  time  tables  are  used,  especially  such  as  are 
adopted  for  periodical  settlements,  as  those  in  banking, 
the  purchases  need  not  be  arranged  in  the  order  of  date, 
as  above. 

16.  Find  the  average  date  of  payment  for  the  following: 

Feb.    3,  Mdse,  3  mos,  $678.59  July  27,  Mdse,  30  ds,  $1500.00 

Mar.   9,      «      2     "  243.75  Aug.   9,      "      90  "  175.50 

Apr.  13,      "      4     »  1000.00  Oct.     3,      «        2  mos,  1673.13 

Jun.  17,      «    30  ds,  976.54  Nov.  18,      "        3     "  987.65 

19,      «      3  mos,  786.15  Dec.  13,      «        3     «  685.18 

*Days  of  grace  are  not  allowed  on  invoices. 


AVERAGE. 


185 


182.  'When  cash  goods  are  sold  with  others  on  credit. 

Groods  are  sometimes  classed  as  cash  or  time.  Those 
designated  cash  do  not  always  realize  present  payment, 
any  time  within  a  month  being  considered  cash;  and  even 
at  the  end  of  a  month  cash  has  not  been  exacted,  the  un- 
derstandino-  being  that  a  debt  contracted  on  cash  terms 

o  o 

draws  interest  from  date.     When  computing  average,  cash 
bills  are  considered  due  on  the  day  of  purchase. 


17 


Jan.  1, 
Feb.  6, 
Mar.  8, 
Apr.  17, 


900 X  59=r  53100 
800X125=100000 
700  X  66=  46200 
600X106=  63600 


3000 


JOHN  MICKLEBOROUGH. 

To  Mdse  on  2  months $900.flD 

"       "       "3       " . .  800.00 

"       "       "    cash 700.00 

"       «       "      "    600.00 

EXPLANATION. — 2  months  from  Janu- 
ary 1  is  March  1,  or  59  days;  3  months 
from  February  6  is  May  6,  or  125  days 
from  January  1 ;  March  8  is  66  days  from 
January  1,  and  April  17  is  106  days  from 
January  1.  Hence,  March  30  is  the  aver- 
age  date  of  maturity  or  payment. 

88  days  after  Jan.  1  or  Mar.  30. 


)262900 


Find  the  average  date  of  maturity  of  the  following: 
D.  E.  SOMERSET.  A.  E.  NELSON. 


18 

Jan.  1,  On  3  mos,  $600.00 

Feb.  3,  For  cash,  670.00 

Mar.  3,  6n  6  mos,  950.00 

May  3,  For  cash,  550.00 


July  1,  On  3  mos, 
13,  On  2  mos, 
19,  For  cash, 
23,  On  5  mos, 


19 

$675.00 
619.54 
147.67 

678.44 


20  Sept.     3,  At  30  days,  $937.15 

9,  At  90  days,     897.78 

17,  Cash,  619.18 

Oct.      3,  At  60  days,     777.00 

16 


186          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Required  the  amount  due  on  each  of  the  following  on 
July  1 : 

21.  Jan.  9,  $678.44  at  60  days;  20th,  $419.88  at  cash 
price;  29th,  $789.14  at  3  mos. 

22.  April  9,  $1678  on  3  mos;  June  18,  $1000  at  cash 
price;  21st,  $879.55  on  60  days;  23d,  $371.19  cash;  29th, 
$785.25  cash. 

183.  Average  Applied  to  statements  when  there  are  credits. 

Dr.  C.  A.  WALWORTH.  Or. 

1867.  1867. 

Jan.  1,  To  Mdse,  $300.00     Feb.  9,  By  Mdse,  $200.00 

Mar.  3,    «        "  500.00           23,     u       "  100.00 

300X  0  200X39=7800 

500=61=30500  Dr.  products.      100x53=5300 
gjJo  13100  Or.  products.       ^          ^^ 

300     500)17400 
Bal.  due  500  34|-  or  35  days  from  Jan.  1  or  Feb.  5. 

EXPLANATION. — Assuming  both  purchases  and  sales  to  be  due  on 
January  1,  W.  would  be  entitled  to  a  discount  on  his  purchases 
equal  to  that  on  $1  for  30500  days,  and  I  would  be  entitled  to  a 
discount  on  my  purchases  equal  to  that  on  $1  for  13100  days,  mak- 
ing a  difference  of  17400  days  in  W.'s  favor  on  the  balance,  $500. 
The  discount  on  $1  for  17400  days  is  equal  to  the  discount  on  $500 
for  17400  days^-500r=34|  days  from  January  1.  Whether  to  bo 
reckoned  backward  or  forward  is  easily  determined.  To  be  in  his 
favor,  the  time  for  payment  must  be  reckoned  forward;  otherwise, 
it  would  be  reckoned  backward. 

Required  the  equated  time  of  the  following: 

WILLIAM  SILLETS. 

24 

Feb.    3,  To  Mdse,   $250.00     Mar.  2,  By  Cash,       $300.00 
Mar.    9,    "       "          300.00     Apr.  3,    "       "  200.00 

Apr.  18,    «       «          500.00 


AVERAGE.  187 

Dr.  J.  C.  HINTZ.                                Or. 

1869.  1869. 

July  3,  To  Mdse,  $1000.00     Aug.    1,  Bj  Cash,   $500.00 

7,    "       "  500.00              13,    "       «         500.00 

Au.  18,    "       "  250.00 

Assuming  the  day  of  settlement  to  be  July  1,  we  have 

1000  X   2=  2000  Days.    500x31=15500  Days. 

500X   6=  3000     "        500x43=21500     " 
J60X48=12000     «      i^T          3WM  Or.  products. 
1750  17000     "  17000  Dr.  products. 

1000  20000  Difference  of  do. 

$750  Balance  due. 

750)20000(26  ds.       EXPLANATION. — The  sum  of  the  credit  pro- 

150  ducts    being  greater  than  that  of   the    debit 

~  products,  shows   that  the  discount  is  in  my 

favor.     Hence,  in  order  to  settle  on  the  assumed 
4'~){) 
J date,  his  payments  would  be  at  a  discount  of 

50  26  days  for  the  balance;  but  as  it  would  be 

impossible  to  settle  on  a  past  date,  we  will  have  to  charge  him 
interest  from  20  days  prior  to  July  1  (June  5)  to  the  real  day  of 
settlement,  whatever  that  may  be,  or  else  take  his  note,  dated  June 
6,  bearing  interest  from  date.  Say  the  date  of  the  settlement  is 
September  1.  Interest  on  $750  from  June  5  to  September  1  is  $11, 
which,  added  to  $750— $761,  balance  due. 

This  balance  is  9  cents  less  than  what  would  be  obtained 
by  computing  interest  on  both  sides  of  the  account  to  Sep- 
tember 1,  and  is  caused  by  ignoring  the  T^  of  a  day — 26 
days  being  taken  instead  of  26^. 

NOTE. — Book-keepers  sometimes  omit  the  tens  of  dollars  when 
averaging.  In  the  above  example  the  hundreds  might  have  been 
omitted  without  serious  error. 


188          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Dr.  WILLIAM  P.  WALLACE.  Or. 

1869.  1869. 

Feb.  22,  To  Mdse.    $500.00     Apr.     3,  By  Cash,  $620.00 

Mar.  28,    "       "  700.00     July     6,    "       "  520.00 

Apr.  30,    "       "  900.00     Sep.   10,    "       "  900.00 

Juii.    8,    "       "  600.00     Dec.     1,    «       «  650.00 

Assuming  January  1  as  the  day  of  settlement,  we  have 
the  following  formula: 

500X  52=  26000  620X   92=  57040 

700X   86=  60200  520x186=  96720 

900x119=107100  900x252=226800 

600X158=  94800  650x334=217100 


2700  288100  2690  597660 

2690  288100 

10  10)309560 

30956  days, 

Or  84  years,  9  months  and  5  days  (allowing  for  leap-years} 

to  be   counted   backward,  because  the  discount   is  in  my 
*  j 

favor. 

184.  Hence,  when  the  balance  of  the  account  and  tho 
balance  of  the  discount  are  both  in  my  favor,  I  coun 
backward;  when  the  former  only,  I  count  forward,  ano 
vice  versa. 

29.  Goods  bought  on  January  9,  at  60  days,  $1376;  on 
13th,  $780;  on  May  3,  $3400.     Payments    made  May  3 
$1200;  June  8,  $3500;  equated  time  required. 

30.  Balance  of  last  account  brought  down,  including  in: 
terest  to  July  1 ;    goods  bought  August  9,  at  3  months 
$2300;  September  3,  $1500;  December  9,  $500;  and  pay 
ments,  October  3,   $3000;    November   9,  $2500;    averag. 
date  of  payment  required. 

31  to  33.  Find  the  equated  time  of  the  following : 


AVERAGE. 


189 


1869. 

July    3,  To  Mdse,  6  mos, 
15,  "       "       3    " 
u       3    tt 

:<      Cash, 


H.  H.  SCHULTZ. 

1869. 


An.  21, 

Sep.  18, 

Oct.  15, 

21, 

27, 

31, 

Nov.  28, 
30, 


Cash, 

Mdse,  3  mos, 
u      4    u 


560.87 
149.50 
2000.00 
396.40 
175.20 
425.16 
100.00 
506.18 
197.45 
321.16 


July     1,  By  Balance,          127.15 

30,    "    Accept.  60  ds,  300.00 

Aug.  29,    «   Cash,  460.00 

Oct.  20,   "  Note,  3  mos,  1000.00 


31, 
31, 

Nov.  30, 
30, 


Cash 

Mdse  Ret., 
Cash, 
Balance, 


100.00 

250.00 

450.00 

2144.77 


4831.92 


Dec.    1, 

1867. 

To 

Bal., 

2144.77 

THEODORE  LILIENTHAL. 

1867. 

Jan. 

t, 

To 

Balance, 

650.00 

Jan. 

8, 

By 

Mdse,  3  mos 

160.00 

Feb. 

3, 

u 

Cash, 

245.00 

15, 

(t 

it 

6    " 

710.87 

15, 

it 

Note, 

60  ds 

416.87 

Feb. 

H 

u 

u 

2    « 

910.14 

Mar.18, 

it 

Accept,  30  ds,  1000.00 

Apr. 

16, 

It 

11 

Cash, 

1000.00 

Jun. 

4, 

it 

a 

60  " 

750.14 

June 

8, 

11 

It 

4  mos, 

900.00 

16, 

it 

Note, 

3  mos, 

987.64 

15, 

tl 

tl 

4    « 

2500.00 

30, 

tt 

Cash, 

500.00 

17, 

tt 

It 

6    " 

1215.00 

30, 

tt 

Mdse, 

abate. 

200.00 

30, 

tl 

Sunds 

i 

700.00 

JULIUS 

WISE. 

1869. 

1869. 

July 

3, 

To 

Balance, 

1500.00 

Aug. 

3, 

By 

Cash, 

1000.00 

18, 

tt 

Mdse 

,  4  mos, 

750.40 

Sept. 

7, 

U 

Acctc 

.  60  ds, 

500.00 

Aug. 

29, 

it 

it 

4    « 

128.80 

Nov. 

5, 

11 

it 

60    " 

750.00 

Sep. 

30, 

tt 

tt 

4    " 

916.84 

Dec. 

14, 

tt 

Cash, 

2000.00 

Oct. 

10, 

u 

it 

3    « 

500.00 

30,    "    Cash,  675.14 

Nov.  18,   "   Sunds,  564.18 

These  exercises  may  be  omitted  until  the  learner  has 
studied  Exchange. 

188.  When  payments  are  made  before  a  note  or  bill  is 
due,  to  find  how  long  after  maturity  it  should  run  to  bal- 
ance the  interest  on  the  advanced  payments. 


190          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

34.  A    merchant  holds   a   note   of   $500   at    6    months. 
Three  months  before  it  is  due,  he  receives  $100,  and  one 
month  before  it  is  due,  he  receives  $300  ;  how  long  should 
he  allow  the  balance  to  run  to  equal  the  interest  on  the 
advance? 

The  int.  on  §100  for  3  mos=int.  on  $1  for  300  mos. 
The  int.  on     300  for  1  mo  r^int.  on     1  for  300  mos. 

600  mos. 

Hence,  the  interest  on  the  advance  payments  is  equal  to 
the  interest  of  $1  for  GOO  mouths ;  that  is,  a  balance  of 
$1  should  have  run  600  months,  but  the  balance  due  on 
the  note  is  §100;  therefore,  it  should  run  |-{^  months— 6 
mouths. 

PROOF.—  The  int.  of  the  $100  for  6  mos=$ 3. 

The  int.  of  $100  (the  first  pay't)  for  the  3  mos=$1.50 
The  int.  of    300  (the  sec.    pay't)  for  the  1  mo  =  1.50 

Total  interest  on  advance=$3.00 

35.  A  note  of  $600  was  given  January  3,  1868,  payable 
in  6  months.     4  months  before  it  was  due,  $100  was  p;iid 
on    it,  and  3  mouths   before  it  was    due,  $200   was   paid; 
how  long  in  equity  should  the  balance  run? 

36.  A  merchant  owes  $700,  due  8  months  from  the  time 
he  contracted  the  bill ;  5  months  afterward,  he  pays  $200, 
and   two   months  after   that,   $300;    how   long    should   the. 
balance  remain  unpaid? 

37.  If  I  borrow  $600  from  A  at  one  time,  and  $500  at 
another,  each  for  4  months,  how  long  should  I  lend  him 
$1000  to  return  the  favor? 

38.  I  owe  $400,  payable  in  10  months;  at  the  end  of  4 
months  I  pay  $100;  3  months  after  that,  $50;  how  long 
after  the  expiration  of  10  months  may  the  balance  remain 
unpaid  ? 


AVERAGE.  191 

39.  A  owes  $1000  due  in  6  months;  5  months  before  it 
is  due  he  pays  $200,  and  3  months  before  it  is  due  he 
pays  §300 ;  how  long  after  the  expiration  of  the  6  months 
may  the  balance  remain  unpaid? 

187.  Average  applied  to  account  sales. 

An  account  sales  is  a  detailed  statement  of  goods  re- 
ceived by  a  commission  merchant,  and  sold  on  account  of 
another.  The  person  who  sends  the  goods  is  called  the 
shipper  or  consignor;  the  person  to  whom  they  are  sent, 
the  consignee,  and  the  goods,  the  consignment. 

The  duty  of  an  agent  or  commission  merchant  is  to  pro- 
cure the  best  intelligence  of  the  state  of  trade  at  the  place 
where  he  does  business,  including  the  quality  and  quan- 
tity of  goods  in  the  market,  their  present  prices,  and  the 
probability  of  their  rising  or  falling;  to  pay  exact  obedi- 
ence to  the  orders  of  his  employers;  to  consult  their  ad- 
vantage in  matters  left  to  his  discretion;  to  execute  their 
business  with  all  the  dispatch  circumstances  will  admit; 
to  be  early  in  his  intelligence,  distinct  and  correct  in  his 
accounts,  and  punctual  in  his  correspondence. 

188.  An   account   sales    should   state   from   whom    the 
goods  were  received,  or  on  whose  account  and  risk  they 
were  sold,  the  dates  and  terms  of  sales,  the  cash  paid  for 
freight,  drayage,  etc.,  and  the  various  charges,  such  as  in- 

i  suraiice,  commission,  cooperage,  etc.  The  total  amount 
received  will  appear  on  the  right  of  the  account,  and  the 
charges,  etc.,  on  the  left,  or  they  may  be  arranged  as  in 
the  example. 

The  difference  between  the  two  sides  is  called  the  pro- 
ceeds. 

Advances  made  on  goods  are  charged  to  the  shipper's 
account,  not  in  the  account  sales.  Where  goods  are  sold 
promptly  for  cash,  or  on  short  time,  the  account  sales  is 
not  averaged. 


192          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

COMMISSION  HOUSE  OF  STRAIGHT,  DEMING  &  CO. 

Shipment  18.  No.  7828. 

Sales  for  account  of  Messrs.  Gaff  &  Baldwin. 

By  sundries, 
June  4,  T.  B.  Colgan  &  Co.  @  60  days,  8  hhds.  Sugar. 

1095     1020 

1100     1120 

1080     1240     8965 

1200     1110       896     8069     @  ?•&  $600.13 

June  6,  G.  Newton  &  Co.,  @  60  days,  10  hhds.  Sugar. 

1080     1040 

1090     1340 

1120     1020 

1240     1100     11440 

1200     1210       1144     10296     @  6|          8707.85 

June  10,  B.  Vilgers  &  Co.,  @  60  days,  20  hbds  Sugar. 

1060  1240 

1210  1110 

1180  1005 

1055  1285 

1240  1100 

1185  1210 

1300  1325 

1010  1140 

1120  1205     23185 

1205  1000       2318     20867     ©6^         1343.31 

$2651.29 

CHARGES. 

June  1,  P'd  cash  st'r  Landis  for  freight,    $87.18 

«  10,  Dray.  950,  ins.  463,  and  stor.  950,    23.63 

"         Commission  and  guarantee,  132.56        243.37 

Net  proceeds  due  by  equation,  Aug.  13,          $2407.92 

E.  0.  E. 
CINCINNATI,  June  14,  1858. 

STRAIGHT,  DEMING  &  Co., 
Per  F.  JELKE. 


AVERAGE.  193 

Aug.  3,     600X63^37800  June    1,     87 

"      5,     708X65=  46020  "     10,  156X9=1404 

«      9,  1343X69=  92667  ^  ^± 

2651  )176487 

243  1404 

2408  175083(72.6  or  73  days  from  June  1, 

16856         which  gives  August  13. 
6523 

4816 

1707 

2.  Find  the  equated  time  of  payment  of  the  following: 
Account  sales  of  merchandise  sold  on  account  and  risk 

of  Morris  J.  Parry,  New  York: 

Mar.    1,  To  Cash  for  freight,          $50.00  Mar.  15,  By  Cash,  $450.00 

1,   "  Drayuge,                          10.00  Apr.    3,   "  S.  Miner,  75.00 

10,   "  Insurance,                        8.50  May    1,    "  Cash,  31rf.75 

June  20,   "  Storage*  Advertising,  10.00                  7,   "J.Clark.  92.25 

"  Com.  on  $1224  @  2,^,       30.60  Jun.  19,   "  Cash,  288.00 

M.  J.  Parry's  net  pro.,  1114.90 

$1224.00  $1224.00 

3.  Sales  of  100  bbls.  of  molasses  for  acct.   of  C.   H. 
Crane. 

July    3,  F.  M.  Peale,  on  acct.,  20  bbls.,  860  gals...  @  1.25  1075.00 

July    9,  Saml.  A.  Butts,  Jr.,  cash,  15  bbls.,  635  gals.  @,  1.20  762.00 

July  18,  Geo.  T.  Ladd,  cash,  12  bbls.,  495  gals @  1.20  594.00 

Sep.    6,  F.  M.  Peale,  on  acct.,  12  bbls.,  495  gals...  @  1.20  594.00 

Oct.     3,  J.  J.  Marvin,  on  acct.,  31  bbls.,  1285  gals.  @  1.00  1265.00 

Dec.  18,  J.  J.  Marvin,  on  acct.,  10  bbls.,  400  gals..  @    .90  360.00 

$4650.00 
CHARGES. 

July    1,  Cash  paid  freight 58.00 

Drayage 18.00 

Dec.  30,  Cooperage 3.20 

Com.  and  guar.  4  percent 186.00 

C.  H.  Crane's  net  proceeds 4384.80 

$4650.00 
17 


194          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

SALES  FOR  ACCT.   OF  R.  H.  LANGDALE. 
Sept.  3,  G.  F.  Sands,  90  days, 

18  bbls.  Clo.  S.  3120  Ibs  net=52  bus.  @  $6.50  338.00 
Oct.    9,  C.  F.  Howe,  on  60  ds.  note, 

5  bbls.  Clo.  S.  930  Ibs.,  15J  bus...  @  10.00  155.00 
Dec.  5,  G.  A.  Voige,  30  ds.  note, 

10  bbls,  —  2220  —  37 @    8.00  296.00    789.00 

CHARGES. 

Aug.  15,  Cash  pd.  fr't 12.00 

Drayage  on  acct 6.00 

Sept.     1,  Advertising 8.55 

Dec.    30,  Storage 10.00 

Com.  and  guar.,  5  per  cent 39.45 

R.  H.  Langdale's  net  proceeds 713.00     799.00 

E.  E. 

CINCINNATI,  Jan.  1,  1867.  NELSON,  NEPHEW  £  Co. 

Answers:  3  mos.  24  ds.;  3  mos.  10  ds. ;  3  mos. ;  Deo. 
31,  1869;  4  mos.  15  ds.;  July  15,  1869;  Dec.  15,  1867; 
$1919.85;  4  mos.  12  ds. ;  Oct.  17;  March  25;  $4707.34; 
Nov.  2;  49  ds. ;  Feb.  21;  July  11;  April  30;  Dec.  23; 
143  ds. ;  April  27 ;  Dec.  6 ;  Oct.  8 ;  May  17 ;  Dec.  21 ; 
Aug.  24;  June  2;  Aug.  16,  1867. 

ACCOUNTS  CURRENT. 

An  Account  Current  is  a  statement  of  the  entire  trans- 
actions between  two  parties,  generally  for  three  or  six 
months.  It  exhibits  the  whole  sums  given  and  received, 
the  interest  due  on  each  at  the  date  of  the  account,  and 
to  whom  the  balance  of  interest  an9  principal  is  due. 

Without  a  knowledge  of  book-keeping,  an  account  cur- 
rent or  account  sales  can  not  be  fully  understood. 

The  name  of  the  party  against  whom  the  account  cur- 
rent is  made  out  is  always  written  first  and  on  the  left, 
while  the  maker's  name  appears  on  the  right.  When  any 
sums  fall  due  after  the  date  of  the  account,  the  interest  ia 
entered  on  the  opposite  side,  making  it  discount. 


AVERAGE. 


195 


AN  ACCOUNT  CURRENT  WITH  INTEREST  COMPUTED. 

MESSRS.  GRAFF  &  BALDWIN, 
In  Account  Current  and  Interest  Account  with 
Dr.                                                          STRAIGHT,  DEMING  &  Co. 

DATE. 

DKSORIPTION. 

TIME. 

IN'T. 

AMOUNT. 

1867. 
Jan. 
tt 

Feb. 

Mar. 
Apr. 

4 
10 

8 

20 
2 
1.0 
18 

To  Mdse,  as  per  bill  rendered  

Days 
161 
155 

63 
114 
104 
96 
57 
55 
44 
49 

8 

37 
20 
9 
6 

145 
137 

88 

77 

65 
37 
16 

11 

20 

19 
2 
0 
4 
4 
1 
8 

0 

0 
0 
0 
4 

c. 
40 
93 

25 

00 
17 
08 
12 
02. 
10  i 
99 

67i 

03 

92 
74 
15 

17! 

425 
810 

500 
1000 
125 
5 
434 
504 
150 
1101 

500 

5 
275 
495 
4152 
5 
30 

c. 

00 
00 

00 
00 
00 
00 
00 
00 
00 
00 

00 

00 
00 
00 
00 
00 
03 

tt              It             «         U           ((                     « 

"  Your  draft  on  us  at  60  days  sight, 
in  favor  of  II   King  &  Co  

"  Your  sight  draft  on  us  

(t  Mdse,  as  per  bill  rendered    .    ... 

"  Exchange  on  $500  at  1  per  cent.... 
"  Mdse,  as  per  bill  rendered  

tt 
May 

June 

u 

Jan. 
Mar. 
tt 

Apr. 
May 

u  " 

June 

20 

25 
26 
4 

8 

25 
5 

8 
14 

20 

28 
18 

29 
10 

8 
29 
14 

"  Our  draft  on  G.  Wright,  your  favor. 

"  N.  Davis  &  Co.'s  note  at  30  days, 

tf  Discount  on  uncurrent  money  at 
1/1*  Per  cent  

l<  Mdse,  as  per  bill  rendered  

"  Net  proceeds,  sales  100  bbls.  flour. 
"  Mdse,  as  per  bill  rendered  

"  Cash  paid  for  vour  telegram  

"  Interest  in  our  favor  

CR. 

By  draft  on  !New  York,  %  P-  ?••  prem. 
"     Cash  

84 

24 
2 

11 

5 

6 
1 
2 

30 

84 

10515 

03 

29 
28 

74 
17 

76 
23 
67 

03 
17 

1005 
100 

800 
403 

624 
200 
1000 
2407 

3976 
10515 

00 
00 

00 
00 

00 
00 
00 
00 

03 

"     Your    sight    draft    on    Moore   & 
Adams,  our  favor  
"     Draft  on  New  York,  %  Per  cent. 

"     Net  proceeds,  sales  50  bbls.  ino- 

«     Cash  

"     Net  proceeds,  sales  38  hhds.  sugar. 

E   &  0   E 

CINCINNATI,  June  14,  1867. 
STRAIGHT,  DEMING  &  Co., 
per  HILL. 

03 

196          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


XXII.  EXCHANGE. 

189.  IF  A  of  Liverpool  is  indebted  to  B  of  New  York 
in  the  sum  of  $1000   dollars,  and   B  is  indebted  to  C  of 
Liverpool  to  the  same  amount,  it  is  evident  that  the  two 
debts   might   be   paid   without   either   party   sending   real 
money  to  the  other.     A  settlement  could  be  effected  by  B 
sending  the  following  order  to  C : 

$1000.  NEW  YORK,  Jan.  3,  1868. 

At  sight,  pay  to  the  order  of  C  One  thousand  dol- 
lars, value  received,  and  charge  to  my  account. 

To  A,  B 

Liverpool,  England. 

This  method  of  settling  accounts  between  persons  in 
distant  places  is  called  Exchange.  If  between  persons  in 
the  same  country,  it  is  called  Home,  Domestic  or  Inland 
Exchange;  otherwise,  it  is  called  Foreign  Exchange. 

190.  A  Bill  of  Exchange  is  an  order  written  like  the 
one  above ;  it  is  also  called  a  check  or  draft.     The  term 
draft  is  usually  applied  to  inland  bills.     There  are  Sight 
Bills  and  Time  Bills. 

191.  A   Sight  Bill  is   one  payable   at   sight  or  on  de- 
mand, as  the  one  above. 

A  Time  Bill  is  one  that  requires  payment  so  many 
days  after  sight,  or  after  date. 

192.  A  Set  of  Exchange  consists  of  three  copies  of  the 
same  bill  drawn  to  insure  safety  of  transmission,  one  of 
which  being  paid,  the  others  are  void. 

The  person  whose  signature  is  attached  to  a  bill  or 
draft  is  called  the  drawer  or  maker;  the  person  addressed, 
the  drawee;  the  party  to  whom  payment  is  to  be  made, 


EXCHANGE.  197 


the  payee,  and  the  one  who  has  possession  of  it?  the  owner 
or  holder. 

REMARKS  ON  BILLS  AND  DRAFTS. 

1.  Bills  of  exchange,  unless  payable  at  sight  or  on  de- 
mand, require  to  be  " accepted,"  and  should  be  presented 
promptly  to    the   drawee   for    this    purpose.     The    drawer 
then  becomes  the  acceptor,  and  the  bill  is  said  to  be  ac- 
cepted or  honored,  and  is  called  an  acceptance. 

2.  Accepting  a   bill  consists   in   the  drawee  writing  his 
name  across  the  face,  by  which  act  he  becomes  responsible 
for  its  payment.     The  following  is  the  form: 


&C  CO.  CO 


Business  men  prefer  a  draft  to  a  promissory  note,  be- 
cause there  are  three  parties  to  it,  while  the  note,  ordi- 
narily, has  only  two.  It  often  happens,  however,  that 
notes  and  drafts  are  made  payable  to  the  drawers,  which 
leaves  only  one  party  to  the  former,  and  two  to  the  lat- 
ter, though  technically  they  are  considered  the  same  as  if 
drawn  in  favor  of  others. 

When  bankers  receive  unaccepted  bills,  they  send  them 
out  for  acceptance  or  notify  the  drawees. 


198          NELSON'S  COMMON-SCHOOL  ARITHMETIC 

Bills  of  exchange,  like  promissory  notes,  may  be  made 
negotiable  by  the  insertion  of  the  words,  "  to  the  order 
of,"  or  "bearer,"  and  are  subject  to  protest  for  non-pay- 
ment, and  the  indorsement  may  be  special  or  in  blank. 

3.  Bills  of  exchange  can  be  had  at  the  banking  houses 
or  offices  of  exchange  brokers,  and  may  be  drawn  in  favor 
of  the  persons  buying  them,  indorsed  and  mailed  to  their 
creditors;  or  they  may  be  drawn  in  favor  of  some  other 
person,  indorsed  by  him,  and  afterward  indorsed   by  the 
buyer,  and  mailed  to  his  creditors,  as  before. 

4.  When  a  bill  or  draft  costs  neither  more  nor  less  than 
the  amount  of  its  face,  it  is  said  to  be  at  par;  if  less  than 
that  amount,  it  is  at  a  discount;  if  more,  it  is  said  to  com- 
mand a  premium,  and  the  rate  of  discount  or  premium  is 
called  the  rate  of  exchange. 

5.  The  phrase,  "apply  to  my  account,"   or  "your  ac- 
count," "as  advised,"  etc.,  are  not  essential  to  a  bill,  but 
rather  indicate  the  relations  of  the  parties  as  debtors  or 
creditors.     When  the  drawer  is   indebted  to  the  drawee, 
he  would  say,  "  apply  to  my  account,"  but  when  the  drawee 
is  indebted  to  the  drawer,  the  phrase  would  be,  "  apply  to 
your  account,"  or  "put  it  to  your  account."     Should  the 
bill  be  drawn  on  account  of  a  third  person,  he  would  say, 
"put  it  to  the  account  of  A." 

6.  When  the  words  "as  per  advice,"  or  "as  advised," 
are  used,  it  is  presumed  that  a  letter  of  instructions  has 
preceded  the  draft.     In  such  case,  the  drawee  honors  at 
his  risk  in  the  absence  of  such  advice. 

7.  Bills   or   drafts   for  acceptance   must    be    presented 
within  a  reasonable  time.     If  the  drawee  destroy  a  bill  for 
acceptance,  or  refuse  to  return  it  in  twenty-four  hours,  he 

be  deemed  to  have  accepted  it. 


EXCHANGE.  199 

8.  Sight  bills  for  collection  should  not  be  mailed  to  the 
drawee,  as   their   possession    is   presumptive    evidence    of 
payment. 

9.  The    phrase    "value    received"    is    properly    omitted 
when  the  bill  is  drawn  against  funds  of  the  drawer  in  the 
hands  of  the  drawee,  as  is  usually  the  case  with  banking 
houses  when  selling  exchange. 

10.  The   place   of  payment,  separate   from   where  it  is 
drawn,  is  not  usually  inserted  in  a  draft,  unless  an  under- 
standing to  the  contrary  exists  between  the  parties. 

11.  Drafts    are   often   drawn    by   merchants    upon    each 
other   to    raise    money   or  settle   accounts.     A    merchant 
shipping  a  large  quantity  of  goods  to  another  to  sell  on 
commission,  usually  draws  a  draft  for  a  part  of  the  cost 
on  the  party  and  sells  it  at  bank,  or  passes  it  to  another 
merchant  in  the  course  of  business.     This  kind  of  paper 
is  called  a  mercantile  draft,  to  distinguish  it  from  one  is- 
sued by  a  bank,  which  is  called  exchange,  or  a  bank  check 
or  draft,  and   is  not  so  available  for  transmission    as  the 
bank  draft  or  exchange.     It  is  a  part  of  the  business  of  a 
banking  house   or  exchange  office  to  buy  this  mercantile 
paper,  send  it  home  for  collection,  and  in  the  mean  time 
sell  exchange  on  the  banks  to  which  they  transmit  it,  for 
such  sums  as  may  be  demanded. 

DRAFT    OF    A    MERCHANT    UPON    ANOTHER    TO    WHOM    HE 
HAS   SHIPPED   GOODS. 

NEW  YORK,  May  17,  1867.     , 

At  ten  days'  sight,  pay  to  our  order  One  thousand  dol- 
lars, value  received,  and  charge  to  our  account. 

To  HENRY  L.  WEHMER.  A.  J.  RICKOFF  &  Co. 

Cincinnati,  Ohio. 

To  obtain  money  on  this,  Mr.  Rickoff  would  indorse  it 
to  a  banking  house,  which  would  pay  him  the  current 


200          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

rate  for  mercantile  paper,  and  discount  for  the  time  to  ex- 
pire before  collection — say  4  days  to  reach  its  destination, 
and  13  days  for  maturity. 

BILLS  IN  DUPLICATE   AND  SETS  OF  EXCHANGE. 

To  prevent  delay  and  guard  against  loss,  bills  of  ex- 
change  are  often  drawn  in  duplicate  or  in  sets.* 

A  duplicate  bill  is,  as  the  name  indicates,  a  correct  copy 
of  the  original,  with  the  addition  of  the  word  "  Dupli- 
cate" written  or  printed  across  the  face.f 

A  set  of  exchange  properly  belongs  to  Foreign  Ex- 
change, under  which  subject  the  learner  will  find  the  form 
and  description. 

HOME  EXERCISES  IN  DRAWING  DRAFTS. 

1.  Draw  on   your  teacher,  (locating  him   in  San  Fran- 
cisco, California,)  at  10  days'  sight,  for  $2136.50,  and  make 
the  draft  payable  to  yourself. 

2.  At  60  days  after  date,  draw  on  0.  I.  Mitchell   for 
$2000,  favor  of  yourself,  and  prepare  draft  for  negotiation. 

Draw  bills  and  notes  from  the  following  data: 

3.  Draw  on  M.  Garaghan,  St.  Louis,  for  $3600,  at  sight, 
and  prepare  the   draft  for  collection  by  the  Central  Na- 
tional Bank,  Cincinnati. 

*  Should  a  bill  be  lost  in  transmission,  the  amount  can  be  recov- 
ered of  the  bank  from  which  it  was  bought,  unless  it  can  be  proved 
that  payment  was  made  by  the  drawee. 

t  On  bank  drafts  will  often  be  found  writing  on  both  back  and 
face,  which  can  not  be  represented  in  type,  such  as  the  names  of 
bank  officers,  through  whose  hands,  they  pass  before  issue,  the 
amount  written  on  the  back  or  face  a  second  time  to  prevent  alter- 
ation. They  are  also  drawn  payable  in  "gold"  or  "currency,"  as 
occasion  requires. 


EXCHANGE. 


201 


Drawer, 

Drawee, 

Payee, 

Date, 

Where, 

Time, 

Amount, 


4.  5. 

Yourself.  Drawer,    Yourself. 

Your  teacher,  at  N.  0.  Drawee,    W.  A.Fillmore,N.Y. 


G.  A.  Carnahan.  Date, 

The  present.  Time, 

Cincinnati,          Ohio.  Payee, 


April  1,  1867. 
10  days'  sight. 
Yourself. 


At  sight. 
$367.25. 


Amount,   $5907.84. 


6.  Face,  $3167.85. 

Date,  August  9,  1866. 

Payee,  B.  0.  M.  De  Beck. 

Indorsee,  Luther  W.  Strafer. 

Drawer,  Noble  K.  Royse,  Cincinnati,  Ohio. 

Drawee,  Herman  H.  Raschig,  New  Orleans. 

Time,  10  days'  sight. 

7.    ACCEPTANCE  OF  F.  M.  PEALE  AT  NEW  YORK. 

Drawer,  Wm.  H.  Morgan,  San  Francisco. 

1st  Indorser,  J.  M.  Allen. 

Time,  60  days  after  date. 

Face,  $967.18. 

Date,  24th  May,  1866. 

Indorsee,        Geo.  F.  Sands. 

8.  Drawer,  John  J.  Marvin,  Cincinnati,  Ohio. 

Drawee,  John  S.  Highlands,  Columbus,  Ohio. 

Payee,  E.  H.  Prichard,  Boston,  Mass. 

1st  Indorser,   B.  B.  Stewart,  Face,  $2127. 
Time,  30  days  after  sight. 


Date, 


January  1,  1867. 


Accepted  two  days  afterward. 

9.  Indorsee,  G.  W.  Harper. 

Drawee,  C.  R.  Stuntz,  Washington,  D.  C. 

Drawer,  G.  W.  Smith,  New  York. 

Indorser,  G.  A.  Schmitt. 

Time,  10  days'  sight,  Face,  $5000. 

Date,  January  1,  1867. 
Accepted  four  days  after  date. 


202          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
AN  INDIANA  NOTE. 

$500.  INDIANAPOLIS,  May  25,  1867. 

Six  months  after  date,  I,  the  subscriber,  of  Indianapolis, 
County  of  Marion,  State  of  Indiana,  promise  to  pay  to  the 
order  of  Geo.  W.  Runyan,  Five  hundred  dollars,  without 
any  relief  whatever  from  valuation  or  appraisement  laws. 
Value  received.  Payable  at  First  National  Bank. 

JAMES  W.  WILSON. 

No.  59.     Due  November  28.  1867 


XXIII.  FRACTION'S. 

193.  A  FRACTION  is  a  part  of  one  or  more  things  con- 
sidered as  a  whole,  and  is  therefore  the  result  of  division. 

Whole  numbers  are   sometimes  expressed   in  fractional 
form.     (See  Improper  Fractions.) 

194.  The  subject  of  fractions  is  the  method  of  treating 
fractional  numbers,  or  showing  how  they  may  be  added, 
subtracted,  multiplied  and  divided. 

195.  Fractions  are  divided  into   common  and  decimal, 
according  to  the  way  in  which  they  are  written.     A  com- 
mon fraction  requires   two  numbers   to   express   it,  as  J, 
while  a  decimal  requires  only  one,  with  a  period  at  the 
left,  as  .5. 

The  following  fractions  would  be  read  as  shown  oppo- 
site: 

•jiy     One  Twelfth. 

•fr     Three  Seventeenths. 

Nine  One  hundred  and  forty -fifths. 

Fifty    seven    Three    thousand    three    hundred    and 
ninety -sixths,  or  Thirty-three  hundred,  etc. 


FRACTIONS.  203 

198.  The  two  numbers  forming  a  fraction  are  called 
terms;  the  upper  term  the  numerator,  and  the  lower  term 
the  denominator.  The  line  between  the  terms  is  the  sign 
of  division,  and  indicates  that  the  upper  term  is  divided 
by  the  lower.  (Art.  34.)  -f%  represents  the  twelfth  part 
of  3,  or  3  parts  of  something  divided  into  12  parts. 

A  fraction  also  expresses  the  ratio  between  the  two 
terms.  ^=3 : 12. 

PRINCIPLES  OF  FRACTIONS. 

197.  If  both  terms  of  a  fraction  be  multiplied  by  the 
same   number,  the  value   of  the  fraction  will   remain  un- 
altered. 

Let  the  1  and  4  of  the  fraction  J  be  multiplied  by  2, 
and  we  have  -|,  a  fraction  of  the  same  value  as  J. 

198.  If  both    terms   of  a   fraction   be   divided   by  the 
same  number,  the  value   of  the  fraction  will  remain  un- 
altered. 

Let  3  and  6  of  the  fraction  -§•  be  divided  by  3,  and  we 
have  ^,  a  fraction  of  the  same  value  as  -|. 

199.  If  the  numerator  only  be  multiplied,  the  value  of 
the  fraction  will  be  increased  and  the  whole  fraction  mul- 
tiplied. 

Let  2  of  the  fraction  f  be  multiplied  by  4,  and  we  have 
•|;  that  is,  eight  thirds  instead  of  two  thirds. 

200.  If  the  denominator  only  be  multiplied,  the  value 
of  the  fraction  will  be  decreased,  and  the  whole  fraction 
divided. 

Let  the  3  of  the  fraction  §  be  multiplied  by  2,  and  we 
have  | ;  that  is,  two  sixths  instead  of  two  thirds. 

201.  If  the  numerator  only  be  divided,  the  value  of  the 
fraction  will  be  decreased,  and  the  whole  fraction  divided. 

Let  4  of  the  fraction  |-  be  divided  by  2,  and  we  have 
|-;  that  is,  two  eighths  instead  of  four  eighths. 


204          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

202.  If  the  denominator  only  be  divided,  the  value  of 
the  fraction  will  be  increased,  and  the  whole  fraction  mul- 
tiplied. 

Let  8  of  the  fraction  -|  be  divided  by  2,  and  we  have 
•J  ;  that  is,  two  fourths  instead  of  two  eighths. 

Common  fractions  are  divided  into  simple,  proper,  im- 
proper, contfpvund,  complex  fractions  and  mixed  numbers. 

A  simple  fraction  is  a  single  fraction,  as  f  . 

A  proper  fraction  is  a  single  fraction  whose  numerator 
is  less  than  the  denominator,  as  jr. 

An  improper  fraction  is  a  single  fraction  whose  numer- 
ator is  equal  to  or  greater  than  the  denominator,  as  f,  f  , 
which  indicates  not  a  part,  but  the  whole  or  more  than 
the  whole;  hence,  the  term  improper. 

203.  A  compound  fraction  is  a  fraction  of  a  fraction  or 
fractions,  as  %  of  f  or  4  of  -J-*  of  ||. 

204.  A  complex  fraction  is  one  having  a  fraction  in  the 
numerator  or  denominator,  or  in  both,  as 


205.  A  mixed  number  is  composed   of  a  fraction  and 
whole  number  together,  as  7f  . 

REDUCTION  OF  FRACTIONS 

206.  Fractions  are  often  expressed  in  terms  too  large 
for   convenient  use,  or  to   estimate   their  value   at  sight. 
The   fraction  -^  possesses   the   same   value  as  J,  and  for 
convenience  in  operating  ought  to  be  reduced  to  that  de- 
gree of  simplicity. 

The  process  of  changing  the  form  of  a  fraction  in  this 
manner  is  called  reducing  it. 

207.  To  reduce  a  fraction  to  its  lowest  terms. 


FRACTIONS.  205 

We  divide  both  terms  by  any  number  or  numbers  which 
will  do  so  without  a  remainder.  (Art.  198.) 

1.  9   and   27    of  the  fraction  ^  divided  by  9,  give  1 
and  3,  or  the  fraction  ^. 

When  a  single  number  will  not  reduce  a  fraction  to  its 
lowest  terms,  other  numbers  are  used  and  the  process  con- 
tinued.* 

2.  To  reduce  --  to  its  lowest  terms. 


< 

3  to  13.  Reduce  the  following  fractions  to  their  lowest 
terms: 
H>  4tt.  T¥S,  iWr.  T#¥&.  T&V&.  dMr.  isWff.  If!!. 

9  3  6          9876 
2"8lT8>  TSTTOT- 

Answers  :  |,  |,  yffj-,  ^fff  ,  |,  3\,  |,  f  f  ,  •#&»  i»  dhr- 

208.  ^  raise  a  fraction  to  a  higher  denomination,  we 
multiply  both  terms  by  the  same  number  —  a  process  the 
reverse  of  the  last. 

209.  When  the  higher  denomination  is  given,  the  mul- 
tiplier may  be  obtained  by  dividing  the  new  denominator 
by  the  old. 

*THE  GREATEST  COMMON  DIVISOR 

Is  the  greatest  number  which  will  divide  any  two  or  more  numbers. 
It  may  be  found  by  the  following  process,  but  the  operation  is  so 
long  that  it  is  seldom  used  in  practice. 

To  find  the  greatest  common  divisor  of  540  and  612. 
640)612(1  EXPLANATION.  —  The  smaller  number  is   di- 

540  vided  into  the  larger,  and  the  remainder  (72) 

72)540(7  into  the  first  divisor;   then  the  next  remainder 

504  (36)  into  the  last  divisor,  etc. 

36)72(2  The  last  divisor  is  the  greatest  common  di- 

7^          visor,  viz.,  36.     That  is,  no  number  higher  than 
36  will  divide  both  without  a  remainder. 


206          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

14.  To  raise  ^  to  24ths,  we  divide  24  by  6,  which  gives 
4  as  a  multiplier.  5x4—20  and  6x4=24,  making  |f. 

15  to  21.  Raise  f  to  12ths,  $  to  IGths,  -f  to  49ths,  |  to 
28ths,  |  to  120ths,  -^  to  ISOths,  ff  to  500ths. 

210.  To  reduce  a  mixed  or  whole  number  to  an  improper 
fraction. 

22.  Reduce  5|-  to  an  improper  fraction ;  that  is,  in  5-J- 
how  many  eighths? 

5?  EXPLANATION. — In  every  whole  number  there  are  8  eighths, 
8  and  in  5  whole  numbers  there  are  8  times  5,  or  40  eighths, 
"TT  to  which  add  7  eighths  and  the  result  is  47  eighths. 

-r 

23  to  30.  Reduce    the   following   numbers    to    improper 
fractions:  15f,  7f  6$,  51f,  17£,  113ft,  ™U,  21f 
Answers :  -*f<s  fy  M,  ^,  *f£>  Aif>  ¥«S  ^^  ¥ 

211.  ^o  reduce  improper  fractions    to  whole    or  mixed 
numbers,  is  an  operation  the  reverse  of  the  last. 

31.     Reduce  ^  to  a  mixed  number. 
8)47 

~&i 

32  to  41.  Reduce  the  following  to  whole  or  mixed  num- 
bers, and  the  remaining  fractions  to  their  lowest  terms: 

•"•t4.  H*.  -f  A>  W.  Hf6->  -xr.  -H6-0-.  ^iW,  WA  H?-0-- 

Answers:  182,  10ft,  2%  50^,  12^  693^,  17#V. 


XXIV.  DECIMALS. 

212.  A  DECIMAL  FRACTION  expresses  its  value  in  one 
term,  and  is  known  from  a  whole  number  by  its  having  a 
period,  called  a  decimal  point,  at  the  left.  .5  is  a  decimal. 

The  value  of  a  decimal  is  more  easily  ascertained  than 


DECIMALS.  207 

that  of  a  common  fraction,  while  operations  in  decimals 
are  performed  with  nearly  the  same  ease  as  those  in  whole 
numbers. 

213.  Figures  increase  in  a  tenfold  ratio  as  they  are  re- 
moved  one   place   to   the   left,  and  decrease  in   the    same 
ratio  as  they  are  removed  one  place  to  the  right. 

In  the  number  .5  the  figure  is  one  place  to  the  right 
of  the  unit  figure,  and  therefore  possesses  only  one-tenth 
of  the  value  it  would  in  that  place.  In  other  words,  it 
represents  tenths  instead  of  units.  One  place  further  it 
would  represent  hundredths,  as  .05,  and  one  place  further, 
thousandths,  as  .005.  As  common  fractions  these  would 
appear  thus  :  ^,  yjfo  Tf^  ;  hence, 

214.  To  reduce  a  decimal  to   a  common  fraction,   we 
erase  the  decimal  point,  and  write  for  the  denominator  as 
many  ciphers  as  there  are  figures  in  the  numerator,  and 
prefix   the   figure   1.     .075    would   be   written  yj-g-g-   and 
•0047,  TDVw- 

Ciphers  on  the  extreme  right  of  a  decimal  possess  no 
value.  .500  expresses  the  same  value  as  .5,  the  first  be- 
ing -j^nnr,  which,  reduced  to  tenths,  is 


1  to  21.  Find  the  fractional  value  of  the  following  and 
read  them:  .23,  .007,  .013,  .760,  .00019,  .3401,  .67800, 
.0907,  .0076,  .3467,  .1093,  .0770,  .3657,  .2136,  .09876, 
.000001,  .13607,  .06789,  .03146,  .000016,  .016037. 

215.  As  in  Federal  money,  the  removal  of  the  decimaJ 
point  one  place  to  the  right  multiplies  a  number  by  10,  and 
its  removal  one  place  to  the  left  divides  it  by  10. 

Removed  to  the  right  in  1673.27,  we  have  16732.7,  and 
removed  to  the  left  we  have  167.327. 

The  notation  and  numeration  of  decimals  being  so  sim- 
ilar to  that  of  whole  numbers,  little  trouble  will  be  expe- 
rienced in  reading  the  following: 


208          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


22.  .01 

28.  .3013 

34.  .45689 

40.  .00001 

23.  .57 

29.  .0031 

35.  .18654 

41.  .0010906 

24.  .709 

30.  .2160 

36.  .36109 

42.  .3016031 

25.  .856 

31.  .1061 

37.  .00009 

43.  .0016039 

26.  .2913 

32.  .4064 

38.  .01002 

44.  .0000067 

27.  .0016         33.  .5067          39.  .168002       45.  .00001001 

Express  in  figures  the  following: 

1.  One  tenth. 
-  2.  Three  hundredths. 

3.  Five  thousandths. 

4.  Sixty-five  ten  thousandths. 

5.  Three  hundred  seventy-six  thousandths. 

6.  Four  hundred  ten  thousandths. 

7.  Eighteen  hundred  and  twenty  thousandths. 

8.  Sixty-one fand  eighteen  hundredths. 

9.  Forty-five  hundredths. 

10.  Eighty-seven  thousand  jand  sixty  hundredths. 

11.  Five  hundred  thousand  and  seven  tenths. 

12.  One  hundred  and  one  thousand  and  seven. 

13.  Sixty-four  thousand  and  eight  tenths. 

14.  Nine  millions. and  seventy-nine  thousandths. 

15.  Eighty-six  hundredths. 

16.  Seven  thousand  and  six  hundredths. 

17.  One  hundred  and  ten  and  sixty-five  hundredths. 

18.  Eighteen  hundred  and  sixty-seven^and  seventy-five 
hundredths. 

19.  Twenty-four   hundred    and    five   hundred   and    one 
thousandths. 

ADDITION  OF  DECIMALS. 

216.  When  arranged,  tenths  under  tenths,  hundredths 
under  hundredths,  etc.,  decimals  are  added  and  subtracted 
precisely  as  whole  numbers.  The  operations  in  Federal 


DECIMALS.  209 

money,  with  which  the  learner  is  already  familiar,  properly 
belong  to  this  subject.  In  those,  the  decimal  points  were 
placed  directly  under  each  other.  The  same  rule  should 
be  observed  in  adding  or  subtracting  decimals  generally. 

I.  To  add  1.07+.001+37.045+10.06+.0007. 

1.07  EXPLANATION. — Here  the  decimal  points  are  arranged 

.001        directly  under  each  other  and  addition  performed  as  in 

37.045        whole  numbers. 

10.06 
.0007 

48.1767 

2.  2.13     +     .426  +  21.2       +     7.63     +  640.072=? 

3.  43.27     +  9.042  +712.417  +  41.007  +         .962=? 

4.  820.71     +  2.006  +  84.243  +217.072  +       9.341=? 

5.  107.67     +  1.301  +  20.0163+684.6       +     10.06  =? 

6.  719.86     +     .2103+       .1610+310.6       +2134.       =? 

7.  9.8784+29.8       +  67.19     +     7.916  +  379.       =? 

8.  643.72     +     .109  +360.06     +       .0006+         .216=? 
Answers:     2748.3678,     671.458,     1133.372,     823.6473, 

806.698,  1004.1056,  493.7844,  2748.3678,  2708.3768, 
493.7844,  3164.8313. 

9.  .007+31.06     +     .1009+100.07    =? 

10.  710.34  +  2.406  +67.709  +       .0006=? 

II.  314.60  +     .0006+     .0027+       .001  =? 
12.  714.06  +     .003  +  8.007  +800         =? 

Answers:    314.6043,    1522.07,    131.2379,    780.4556, 
1522.3074. 

SUBTRACTION  OF  DECIMALS. 

When  the  larger  number  has  fewer  places  of  decimals 
than  the  smaller,  the  blanks  may  be  filled  with  ciphers. 
(Art.  214.) 
18 


210          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

1.  To  find  the  difference  between  107.06  and  .213. 

OPERATION.     107.060 
.213 

106.847 

2.  617.07—  41.7106:=?  6.  341.     —.213  =? 

3.  10.06—     .9092=?  7.         .97—  .0376=? 

4.  36.84—  6.672  =?  8.       4.15—  .1999=? 

5.  118.09—  7.009  =?  9.       7.96—  .9789=? 
Totals,  351.6056,  725.7592,  352.6506. 

MULTIPLICATION  OF  DECIMALS. 


217.  In  this  rule  we  multiply  as  in  whole  numbers, 
and  mark  off  as  many  places  of  decimals  in  the  product 
as  there  are  in  the  two  factors. 

1.  To  multiply  6.107  by  5.5. 

6.107     There  are  three  places  of  decimals  in  this  factor, 
5.5     and  one  place  in  this; 

30535 
30535 


33.5885     so  we  point  off  four  in  the  product, 
PROOF.     6.107X  5=30.535 
6.107X.5=  3.0535 
33.5885 

EXPLANATION. — The  6  and  107  thousandths  multiplied  by  6—30 
and  535  thousandths,  but  multiplied  by  .5  or  5  tenths,  it  is  only  one 
tenth  as  much,  or  3.0535  (Art.  ^13),  which,  added  to  the  first  product, 
gives  33.5885,  as  above. 

2.  .3507 X  10.09  =?  7.  2300.7     X  48.003  =? 

3.  17.07     X200.6     =?  8.     704.23  X       .0007=? 

4.  785.4       X  36.70  =?  9.  .786x100         =? 

5.  .279  X160.7     =?          10.         4.862X       .75     =? 

6.  876.5       X       .780=?          11.     200.03  X       .002  =? 

Total,  32980.4659.  Total,  110523.641621. 


DECIMALS  211 

218.  When  the  product  contains  fewer  figures  than  there 
are  decimals  in  the  factors,  the  number  is  made  up  by  pre- 
fixing ciphers. 

12.  100X.0005=? 

100 
.0005 

500,  to  which  prefix  one  cipher  and  we  have  .0500,  or 
.05,  the  answer. 

PROOF.     .0005X100=.05. 

13.  .107  X-05     ==?  16.  .3045X.00061=? 

14.  61.04     X-0007=?  17.  .27     X-27       =? 

15.  .7103X-004  ==?  18.  .4102X-1004  =? 
Totals,  .0509192  and  .114269825. 

DIVISION  OF  DECIMALS. 

219.  When  dividing  decimals,  the  quotient  and  divisor 
must  contain  as  many  places  of  decimals  as  the  dividend. 

1.  33.5885-=-6.107=r? 

6.107)33.5885(5.5          PROOF.—  This  is  the  converse  of  Ex.  1  in 
30  535  multiplication,  the  multiplier  and  multipli- 

cand  being  6.107  and  5.5  and  the  product 

_ 


30535 

A  further  proof  is  obtained  by  estimate,  if  we  divide  the  whole 
number  (83)  of  the  dividend  by  the  whole  number  (6)  of  the  divisor, 
which  will  give  one  pla-ce  for  whole  number. 

220.  When  the  dividend  does  not  contain  as  many 
decimals  as  the  divisor,  ciphers  may  be  annexed  to  make 
up  the  number.  The  quotient  will  then  be  a  whole  num- 
ber, as  it  simply  shows  the  number  of  times  the  latter  is 
contained  in  the  former.* 

*In  practice,  decimals  are  seldom  carried  to  more  than  four  places. 


212 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


2.  3066-r-.1783=? 

.1783)3066.0000(17195.73752 


1783 

12830 
12481 

3490 
1783 


17070 
16047 

10230 
8915 

13150 
12481 

6690 
5349 


In  this  example  four  ciphers  have  been  an- 
nexed to  the  dividend,  to  correspond  with  the 
number  of  decimals  in  the  divisor.  From 
this  resulted  17195,  the  quotient.  These  ci- 
phers being  exhausted,  five  more  were  an- 
nexed to  the  remainders  to  give  the  five  deci- 
mals in  the  quotient. 

Another  method  is  to  annex  ciphers  at 
will,  observing  to  place  a  mark  in  the  divi- 
dend to  limit  the  whole  numbers  in  the  quo- 
tient, as  3066.0000000. 


13410 
12481 

9290 
8915 


3750 

3566 

~184 

221.  When  there  are  not  figures  enough  in  the  quotient  to 
make  up  the  number  of  decimals  in  the  dividend,  ciphers 
should  be  prefixed  to  the  former. 

3.  Divide  10.70067  by  370.4. 

370.4)10.  70067(.0288        Here  the  quotient  produced  only  three 
7  408  figures  (288),  which,  with  the  one  in  the 

0  .  divisor,  makes  only  four  decimals,  so  to 

make  UP  tne  number  a  cipher  is  prefixed. 


32947 
29632 


DECIMALS.  213 

Carry  out  the  following  to  only  four  places  of  decimals : 

4.  314.06  ~  10.73     =?          8.       6.74—2.34  ==? 

5.  17600        -f-785.4      =?          9.  496      -*-  .278=? 

6.  3170.09  -r-     2.4014—?        10.       7.6  -5-  .734=:? 

7.  417.456-5-  31.145  ==?        11.       7.23--4.06  =? 
Totals,  $1385.1824  and  1799.1878. 

12.  30.640  -493.67  =  ?      16.  724.1       -f-38.07       =? 

13.  10.8739-^-117.406=?      17.     82.03     -~-  9.0002  =? 

14.  6.342  ~  22.973=?      18.       7.624  -5-  2.001     =? 

15.  1467.06     -=-196.04  =  ?      19.         .5213-7-     .24121=? 
Totals,  7.9142  and  34.10573. 

REDUCTION  OF  DECIMALS. 

222.  To  reduce  a  Common  fraction  to  a  Decimal. 

1.  Reduce  J  to  a  decimal. 

2)1.0         By  annexing  a  decimal  point  and  a  cipher,  the  number 
7       is  properly  reduced  to  tenths  or  10  tenths,  in  which  2  is 
contained  5  times.     This  5,  being  of  the  same  denomi- 
nation of  the  dividend,  is  tenths,  or  .5. 

2.  Reduce  J  to  a  decimal. 

3)1.00000         This  quotient  may  be  carried   out   indefinitely, 

oqo oq      and  ig  called  a  repeating  decimal.     To  save  writing, 

a  point  is  usually  placed  over  the  repeater  thus  .3. 

223.  The  fractional  value  of  a  repeating  decimal  may 
be  restored  by  using  9  instead  of  10  as  the  denominator, 
as  f =f 

3.  Reduce  ^-  to  a  decimal. 
7)1.000000000000 

142857142857+      This  is  called  a  circulating  deci- 
mal and  is  marked  thus :  J42857=|ff||-£=f 

Its  fractional  value  is  restored  in  the  same  manner  as 
that  of  the  .3  in  the  preceding  example. 


214          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Express  the  following  decimally:* 

4.  J          8.  A          I*.  H          16.  A          20. 

5.  i          9.  ff  13.  ^          17..  &          21. 

6.  j         10.  ^          14.  ^  18.  43          22. 

7.  f         11.    J  15.  H          19.  ^         23. 
Answers:    .75,  .4,  .25,   .375,   .1923+,    .6043,   .45,    .16, 

.6875,  .416,  .2187+,  .046875,  .9,  .475,  .00224,  .46,  183, 
.6,  .02,  .5375,  .2. 

224.  To  find  the  value  of  the  decimal  part  of  a  com- 
pound number,  as  £0.75  or  &0.33J. 

24.  What  is  the  value  in  shillings,  pence  and  farthings 
of  £0.345? 

.345  EXPLANATION.  —  Multiplying  .345,   that  is, 

20  fbVu  by  ^'   tlie   number  °f  shillings   in   a 

'  or   6'900 


^       ,  [A^  n^  ^d  muhiplying  the  900  shminga 

by  12,  the  number  of  pence  in  a  shilling,  we 


10.800  Pence  have   Uffffi  or   10.800    pence.     Multiplying 

4  the  .800  by  4,  the  number  of  farthings  in  a 

3.200  Farthings  Penny>  we  have  TSa8  or  3-200  or  3'2  farth' 
or  6s  lOd  34-  far.  inS8- 

The  operation  might  have  been  abbreviated  by  dropping 
the  ciphers  on  the  right. 

Hence,  we  have  for  the  result  6  shillings  10  pence,  3 
farthings  and  T20  or  ^. 

Find  the  value  of 

25.  .625     of  a  gallon.  28.  .1374  of  a  ton. 

26.  .1425  of  a  year.  29.  .0037  of  a  Ib.  Troy. 

27.  .8323  of  a  £. 

Answers:  21T%;  2,  1;  274,  12T»ff;  16,  7f  ;  1,  21TV 
225.   To  reduce  denominate  values  to  decimals. 

*The  plus  sign  will  be  used  when  the  decimal  can  be  carried  out 
further. 


DECIMALS.  .  215 

30.  Reduce  6  shillings  10  pence  3^  farthings  to  the  dec- 
i  nal  of  a  pound  sterling. 

5)    1.0  The  first  step  in  this  operation  is 

j\    o  9             q  <2    /  ^°  reduce    7=    to    a    decimal,    which 

f — —               T'  gives  .2.     Prefixing  3  farthings,  we 

12)10.8  o?*  lO^j-  pence,  divide  by  4,  the  number  of  farthings 

f>(\\   R  a  ™  A  9     cJiV77  ina  penny,  and  obtain  .8  of  a  penny. 

^jU  )       O.5J  Cf™    U^  g     b/illl/.  r  •*  . 

— ^ to  which  we  prefix  10  and  divide  by 

.345  or  i'O.-j^o-  12,  the  number  of  pence  in  a  shil- 
ling and  obtain.  9.  Prefixing  6  shillings,  we  divide  by  20,  the  num- 
ber of  shillings  in  a  pound,  and  obtain  .345  of  a  pound,  the  answer. 
The  reverse  of  this  process  is  found  in  example  24. 

The  pupil  can  prove  his  calculations  by  last  Art. 

31.  Reduce  3  quarters  to  the  decimal  of  a  yard. 

32.  Reduce  6  Ibs.  3  oz.  to  the  decimal  of  a  cwt 

33.  Reduce  12s.  6fd.  to  the  decimal  of  a  £. 

34.  Reduce  12  Ibs.  to  the  decimal  of  a  tun. 

35.  Reduce  1  ft.  3£  in.  to  the  decimal  of  a  yard. 

36.  Reduce  16  oz.  to  the  decimal  of  a  ton. 

37.  At  56  cents  a  pound,  what  will  127  Ibs.  6  oz,  of  tea 
come  to? 

16)60 

.375  decimal  part  of  a  pound. 

127.375 
56 

764  250 
6368  75 

7133  000  cents,  or  $7133. 

REMARK. — This  is  not  strictly  a  practical  question,  nor  the  short- 
est method  of  computing  the  above,  the  object  being  merely  to  show 
the  application  of  decimals. 

38.  At  $5  for  a  pound  sterling,  what  will  be  the  value 
of  £16  8s.  10d.? 


216          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

39  to  41.  What  will  be  the  value  of  the  following  sums 
of  money  at  the  saine  rate?  £167  10s.  3Jd.,  £19  2s.  6d., 
£10  10s.  10£d. 

Answers:  $95.63,  $837.57,  $52.72,  $82.21. 


XXV.  COMMON  FRACTIONS. 
MULTIPLICATION. 

226.  A  fraction  is  multiplied  by  a  whole  number  by 
simply   multiplying   the   numerator   without   altering  the 
denominator.     [Art.  199.] 

|  X  7=7X3,  or  ^-,  which,  reduced  to  a  mixed  number, 
equals  5J. 

227.  Fractions  can  also  be  multiplied  by  dividing  the 
denominator,  without  altering  the  numerator.     [Art.  202.] 

AX5=f$=i,  or  If 

Multiply  the  following  fractions: 

1.  |X  5=1J  4.  AXH=2-357 

2.  |X  4=3£  5.  ^x  9=3.316 

3.  |X  12=8  6.  78TX  6=2.824 

228.  Mixed  numbers  may  be  multiplied  like  compound 
numbers. 

7.  Let  it  be  required  to  multiply  4f  by  7. 

Whole  Nos.  Eighths. 

Illustration.     4  5        EXPLANATION. — Seven  times  5  eighths 

7      equals  35  eighths,  or  4  whole  numbers 

~~^    ; o      anc*  3  eighths.     Seven  times  4=28  and 

on*  4  make  32.     Ans.  321. 

or  32f 

It  will  not  be  necessary  for  the  pupil  to  write  his  work  in  so 
formal  a  manner  as  in  this  illustration. 


FRACTIONS.  217 

8.  Multiply  6£  by  12. 

6-J  Multiplying  7  ninths  by  12,  we  have  84  ninths  or  9 
12  and  3  ninths.  Then  12  times  6  are  72  and  9  make  81, 
£..  3  giving  for  the  answer  81J. 

or  81] 

9.  6J     X  8=?*  13.  35 1  X  9=?     17.  914  f  Xl20=? 

10.  7|     X  7=?     14.  60|-X12=-?     18.     63  £  X  15=? 

11.  8|     X  6=?     15.  45  f  X  8=?     19.  127T\X  20=? 

12.  1,^X12=?     16.  64^X   6=?     20.  110  ^-X  14=? 
Answers:  54,  50f,  13^,  53  j,  109760,  952.5,   387.29, 

2543.636,  1542.3,  316,  730.5,  3S7-&,  365.3. 

229.  To  multiply  one  fraction  by  another,  the  numera- 
tors may  be  multiplied  together  for  a  new  numerator  and 
the  denominators  for  a  new  denominator. 

21.  f  Xf=3r<V  or  f 

Here  f  is  multiplied  by  3,  giving  6  thirds;  but  this  3  being  sev- 
enths, 6  thirds  is  7  times'  too  much.  Multiplying  the  denominator 
[Art.  200]  by  7  gives  JL  or  ^. 

29      2\s  3\s  4\/  5 1  20 1 
«•   sX^Xs-X-g— 3TTF— •£• 

These  operations  might  have  been  abbreviated  by  what 
is  called  cancellation.  In  the  first  example,  for  instance,  -| 
is  to  be  multiplied  by  ^;  that  is,  the  numerator  2  is  to  be 
multiplied  by  the  numerator  3 ;  but  the  2  is  also  to  be  di- 
vided by  3,  for  -|  signifies  that  2  is  to  be  divided  by  3; 
therefore,  since  the  2  is  to  be  multiplied  by  3  and  divided 
by  3,  it  remains  exactly  the  same,  and  the  3  of  the  de- 
nominator is  said  to  cancel  or  make  void  the  3  of  the  nu- 
merator. In  the  following  operations  the  canceled  figures 
will  be  known  by  having  a  line  drawn  through  them. 

*  Many  answers  in  this  chapter  are  given  in  decimals  carried  to 
two,  three,  or  four  places,  as  in  practice.  Occasionally  the  plug 
sign  is  used  to  show  that  the  decimals  may  be  continued;  at  othei 
times  the  last  figure  will  be  found  increased. 


218          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

2     $     2  In  the  second  operation,  the  2  of  the  nu- 

I     7~7  merator  and  the  6  of  the  denominator   are 

uncanceled,    making   f,  which,    reduced    by 

_X-X-X-     dividing  both  by  2,  equals  J. 

$^06         The  2  and  6    might   have   been   canceled 

t     $     £     $     a^s°5  ^  dividing  both  where  they  stood  in 

-X-X-X-     the  question,  as  in  the  third  example,  plac- 

ing only  3  as  a  denominator,  and  1  as  a  nu- 

merator.     1    is    always    to    be    understood 

where  a  number  has  been  canceled. 


8 


23.  i 

1  Some  prefer  arranging  the  terms  of  can- 

1$  0  3     celing   fractions    as    in   the   margin,   with 

15  the    denominator    or  divisor   on   the   left, 

"^  and  the  numerator  on  the  right. 

EXPLANATION.  —  The  first  2  was  canceled  in  the  18,  leaving  9;  the 
24  and  9  were  canceled  by  dividing  both  by  3,  leaving  8  and  3;  the 
74  was  canceled  by  the  second  2. 


24.  iX    -I  X  f         =?  28.     1JX 

25.  fX    *  XifXTV=?  29. 

9K      Is/       8    N/    5   \/   7  ?  30 

^O.   -^X    T5"^34<^2T — 

27.  |XHXfXj=?  31. 

Answers:  3.375, 4.8,  |,    J,   .05,  5.75,  .0041,  .0264+. 

230.  Compound  fractions  are  reduced  to  simple  ones  by 
multiplication.  Let  it  be  required  to  reduce  \  of  §  of  | 
to  a  simple  fraction.  We  know,  by  inspection,  that  one- 
half  off  is  ^,  and  that  \  of  f  is  J3.the  answer. 

By  multiplication: 


By  cancellation:  -XTX-= 
?>    I,  Jl 


FRACTIONS.  219 

32.  £  of  |  of  ^=?  36.     |  of    f  of  9J         =? 

33.  *  of  #  of  3*=?  37.     fof    }'<&&        =  ? 

34.  f  of  %  of  -&=?  38.  4£  of  li  of  £          =? 

35.  £of  Ij-of  £=?  39.     iof    |x*Xli=? 
•      Answers:  7.389,  1.75,  .0068,  .286,  1.5,  -J,  .4286,  .3. 

231.    To  multiply  one  mixed  number  by  another. 
The  mixed  numbers  are  reduced  to  improper  fractions 
and  multiplied  as  in  Art.  229. 
40.  3J  by  5*=? 


41.44X71=?        44.  4T*rx3|=?         47.     Ifx  8£=? 

42.  2JX1£=?         45.  2JX  f=?         48.  72|x62>=? 

43.  3|X2T5T=?        46.  6g-  X2^=?         49.  87-5X16?=? 
Answers  :  29f£,  17^,  l^J,  8||,  2.7,  1^,  17*i  4537.5, 

1458.3,  4.8,  14.583. 

50.  At  11|  cents  a  pound,  what  will  147J  Ibs.  of  coffee 
cost? 

51.  What  will  7^  Ibs.  of  cheese  cost,  at  9J  cents  per 
pound? 

52.  At  12J  cents  a  pound,  what  will  120  Ibs.  of  sugar 
cost? 

53.  What  will  14^  Ibs.  of  beef  cost,  at  6j  cents  a  pound? 

54.  Fifteen  and  a  half  yards  of  muslin,  at  9^  cents,  will 
cost  how  much? 

Answers:  71  ts,  $14.80,  $16.59,  98  c,  $1.43,  $16.50. 

DIVISION. 

232.  Division  being  the  reverse  of  multiplication,  to 
divide  a  fraction  by  a  whole  number,  we  divide  the  nu- 
merator or  multiply  the  denominator.  [Art.  200—1.] 


220          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
2..  -j-g— 4=?  5.  -»£-±-  7=? 


;.  ^^-9=?- 
.  A-s-6=? 


6.  ^.e^io^?  9. 

4.  J8. 3—?  7.  I8_i_  4—?  10    "7-— 9=? 

36  *    ^Y     *  *    ~§~  1     * 

Answers:  ^,  gL,  ^  .1887,  .076,  .16,  .16,  .125,  f. 

233.  To  divide  mixed  numbers. 
11.  21|-H-6==? 

Whole  NOB.  Fifths.  EXPLANATION. — 6  is  contained  in  21,  3  times 

'_  __  and  3  ^eft"     ln  tne  &  °f  tne  remainder  there  are 

3  3  15  fifths,  which,  added  to  the  3  fifths  in  the  ques- 

or  3-|  tion,  make  18  fifths.  %6  in  18,  3  times.    Ans.  3£. 

12    1 2 '  — —  8 ? 

8)124-  EXPLANATION. — In  this  example  we  had  4  remainder, 

129       in  which  were  28  sevenths,  and  the  one  in  the  question 
&°       made  29.     Then,  as  8  would  not  divide  29  without  a  re- 
mainder, we  multiplied  it  into  the  denominator,  which  made  56. 
Ans.  1||. 

13.  67f-r-  7=?      17.  167^25  =  ?       21.     72f--  9=? 

14.  445 _,_  3==?       is.     21J--14=?       22.  148i-*-27=? 

16.  118^12=?       20.     22J-i-12=?       24.  175f-v-15=? 
Answers:  llf  8^-,  5.5,  9«f,  14||,  6.684,  1.523,  2.306, 
23.05,  11.7142+,  19||,  9£,  1.861. 

234.  To  divide  one  fraction  by  another. 
OR    i  .  3 ? 

ft9»    2~*~?  — 

1     .    3 1  V  4  —  4   ^  2 

•j-s-f =2  X  -3 —^ — B^- 

Multiplying  the  denominator  of  the  dividend  by  3,  we 
have  \  [Art.  200-1],  but  as  the  divisor  is  not  3  but  3 
fourths,  we  multiply  the  result  by  4,  giving  |-,  or  1^-; 
hence,  we  divide  one  fraction  by  another  by  inverting  the 
terms  of  the  divisor,  and  multiplying  as  in  multiplication.* 

235.  To  divide  by  a  mixed  number. 

*  CAUTION.— The  pupil  will  observe  not  to  invert  the  terms  of  the 
number  to  be  divided. 


FRACTIONS.  221 

26.  369--97rr=? 


Hero  both  terms  are  reduced  to  the  same  denomi- 
*™4     obJ     nation  (thirds),  and  division  performed  as  in  whole 

0        ~     numbers.     97J  is  contained  in  369,  3||£  times. 

^292  )H07(3f|i 
876 

231 

27.  76-f-2|  =  ?  30.  349  ~  4J=?  33.  73^8  f  =? 

128.  84-^-3|=?  31.  106f-7-  5|==?  34.  73  £ -=-    ^=? 

^29.  82-r-7f=?  32.  276f-f-12°  =  ?  35.  191 1  --  T\=? 

Ans.      28fft,  23.05,  29.37,  10*f,  77.5,  18 J^,  8.565+, 
719^,  293/254. 

236.    Complex  fractions   are   unsolved  questions   in   di- 
vision. 

36.  2J  0     7 

37.  f-f-f  =?  41.  i  of  f -f-f         =? 

38.  f-^|=?  42.  l|xiH-|         =? 

39.  ^-f- 1  =  ?  43.  2J  X  T-4-|  of  |==  ? 

40.  i-H-i=?  44.  3^      * 

\/  «  .  O     rv-f-     7  •   f 

s\    ~^~    '     >?    ^-*-    "S"~~~ 

45.  If  120-J  Ibs.  of  cheese  cost  $14.80,  what  will  1  Ib. 
cost? 

46.  Find   the  cost   of  1    Ib.   of  coffee,    when   15^  Ibs. 
cost  $1.43. 

47.  If  11^  yards  of  cassimere  cost  $16.59,  what  will  one 
yard  cost? 

48.  If  9-J  yards  of  muslin   cost  71   cents,  what  will  1 
yard  cost? 

Answers:  7^,  9^.,  147/3,  12^r,  1.185,  .6,  .3,  .53,  .36, 
.,    5$,  2.571. 


222  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


SUBTRACTION. 

237.  To  subtract  fractions  or  mixed  numbers  from  wlwle 
numbers. 

1.  From  87  take  25f 

Whole  Nos.   Sevenths.          TO  subtract  3  sevenths,  one  whole  number 

"•  is  added  to   both    terms.     In    1   there   are  7 

sevenths,  from  which  subtract  3  and  the  re- 

61  4  mainder  is  4.     1   to  the  25  makes  26,  which 

take  from  87  and  61  is  left,  giving  for  the  answer  61  A. 

The  above  formula  being  used  for  illustration  only,  the 
learner  will  be  expected  to  write  his  operation  as  follows: 

87 
25* 

~61|~ 

2.  210—371=?     b.  1003—819  *  ==?     10.     29—25  J  =? 

3.  119— 82-J»?    7.  3785—  10  -f- =  ?     11.  167— 89f  =  ? 

4.  61—  4f  =  ?     8.  2168—  14  f  =  ?     12.     36—21^=? 

5.  54_  51=?     9.  1765— 777£|=?     13.  218— 36 1=? 
Answers:  987.4,  2153.571,  3774.14,  172.5,  36.875,   56. 

1,  48.875,  243.8,  172J,  3.25,  77.125,  181.625,  14|f,  3.75. 

238.  To  subtract  a  fraction  from   another  of  the  same 
denomination. 

14.  From  £  take  *. 

6—4=2  or  I 
6  sevenths  less  4  sevenths  leaves  2  sevenths,  the  answer. 

15.    |-i=?  18.  |i-||  =  ?  21. 

16.1-|=?  19.    |_4=?  22. 

17.  i32-TV=?  20.  ^-3\=?  23. 

Answers:  J,  II,  .2,  .25,  .4,  .14285+,  ft,  -i,  .1,  f 

239.  ^b  subtract  one  fraction  from  another,  both  terms 
must  be  of  the  same  denomination,  or  reduced  to  the  same 


FRACTIONS.  223 

denomination.     |  can  be  subtracted  from  f ,  but  J  can  not 
be  conveniently  subtracted  from  |  without  first  changing 
the  denomination  of  one  or  both. 
24,  Subtract  J  from  J. 

EXPLANATION. — The  6,  being  common  to  both  fractions,  is  called 
the  common  denominator.  To  raise  two  fractions  to  a  common  de- 
nominator, both  terms  of  each  fraction  may  be  multiplied  by  the 
denominator  of  the  other.  [Art.  197.] 

oc      3 2  9  ^?ft       4  1  ?  Q"l         3  ___  1  ^ 

rt/>      4^ i    o  on        2  5  y  oo        fj  1   9 

97      7         1  O  Oft       7  4   «>  oo        2  1   ? 

Ail  I  •      TT 15"    4  OV.        "If ?"    — —  *  OO.       7j  y  -|-7j I 

Answers:  |f,  .2576,  .3035+,  .3,  .07,  ^,  .375,  .0396+, 
.075,  .3096. 

ADDITION. 

240.  Fractions  of  the  same  denomination  are  added  to- 
gether  by  finding  the  sum  of  the  numerators. 

1-  l+f+l-2+5+7  thirds,  or  4J. 

s'  A+A+A+i-? 

Answers:  1.336,  .851,  {^J,  J,  17\,  2T4T,  4T\,  4|. 

241.  Fractions    of  different   denominations    are    added 
together  by  finding  a  common  denominator,  as  in  subtrac- 
tion, and  proceeding  as  in  last  Art. 

8.  Find  the  sum  of  f+f+f. 

3X5x6:=90.  Common  denominator. 

1 

\  2X5X6=   60  Here  the  numerators  2,  3  and  5  are  multi- 

'I  3X3X6=    54          plied  successively  by  all  the  denominators  but 

|  5X^X3=   75  their  own.     2,  for  instance,  is  multiplied  by 

-j  oq          &X6}  the  product  of  which  is  also  the  multi- 

I  fpj        .    189 Q  ^     plier   of  the   denominator   3,    giving  for  the 

.    1S  "^°"  -^TTT-  first  fraction  |g  instead  of  J.     In  the  same 
way  |  becomes  |J,  and  |  Jg,  making  189  ninetieths,  or  2.1. 


224          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

When  there  are  many  fractions  to  be  brought  to  the 
same  denomination,  it  will  be  better  to  first  divide  the 
common  denominator  by  each  denominator.  This  gives 
the  number  which  will  raise  both  terms  to  the  required 
denomination.*  [Art.  197.] 

9-    B-  4+   i=2  13.  8  £+6  £  +12    =? 

10.   H-  4+  i=?  14.  2$+  JH-    if=? 

11-      !+   t+    7T=?  I5'      Fr+9      +      TT=? 

12.  6J+7f+8£=?  16.  6f  +lf  +  2£  =  ? 

Answers  :  7J,  10.3714+,  1^,  .4345+,  1.0707,  9.612+, 
32.25,  3.995+,  22.277,  26.75. 


19- 
20. 

21.  |    Of  f  +|   Of 

22.  |XT9ir+  of  fX?  of  £+l+5f  +J=? 

23.  2|X6i+8£+f  of  t+l  of  3+7=? 

24.  lX2+    of     of 


*THE    LEAST  COMMON   MULTIPLE 

Of  several  denominators  is  the  least  number  which  can  be  divided 
by  them  without  a  remainder.  The  following  is  the  process  for 
finding  it: 

To  find  the  least  common  multiple  of  3,  4,  6,  8,  9,  12,  15. 

2)3  4  6  8  9  12  15  EXPLANATION.—  2  was  used  as  a  di- 
3^3  2  3  4  9  6  HT  visor  of  4,  6,  8  and  12,  and  the.  quo- 
277  ~  -  ~  ~  ~  ~  tients  set  down.  The  other  numbers 
were  brought  down  and  3  divided  into 


those  divisible  by  it  without  a  rernain- 


—  36°-  der;  and  so  the  process  was  continued 
until  no  number  could  be  found  to  divide  the  others  without  a  re- 
mainder. The  divisors  and  remaining  numbers  then  being  multi- 
plied together,  produce  360  as  the  least  common  multiple. 

Like  the  operose  method  of  finding  the  greatest  common  divisor, 
this  is  seldom  used. 


FRACTIONS.  225 


25.  4f  x5J+6iX2|  of 

26.  |  of  f  +?  of  I  of 

2T. 


28. 


29.  f  of    f  +  _^+  1  of     ^=1 

30.  4-  +  ^  of  }f  +  |  of  5^=? 

31.  |f  of7f  of  9     +  ~  of  14  ==? 

32.  ££  +  |iof2f  +  |-  of  6f=? 

33.  ^of   i|of9|+^of  8£=? 

34.  »+  «+  ii+e      =^ 

Answers:  14^,  15/T83+,  34.91+,  85.9146,  .01269, 
7.14285,  26.1428,  5.25,  141.85,  175.85,  1.9642,  12  J, 
^W^  3.4531+,  14.183+,  4.350+,  53.83, 
11.168+,  3.0278+,  4.0126+,  32.2678+,  11.694+, 
,  82.61. 


PRACTICAL  QUESTIONS. 

1.  In  an  invoice  of  goods  there  are  the  following  items; 
required  the  amount. 

27  J   doz.  @    9Jc  13f  doz.  @  5^c. 

18T7^     "      "  12  16|     "      «  3J 

16^    «      "  12J  118^-    «      «  2J 

-4ws.  $10.65. 

2.  §  of  a  merchant's  goods  were  destroye^  by  fire,  and 
what  remained  was  worth  $1637.50;  what  was  his  loss? 

3.  A  owns  §  of  a  steamboat,  B  J  and  C  the  remainder, 
which  is  worth  $10000;  what  is  the  value  of  the  boat? 

4.  J  of  a  saw-mill  belongs  to  A,  •§•  to  B,  T3^  to  C,  the 
remainder  to  D,  and  the  profits  for  the  year  amounted  to 
;$1680;  what  is  each  man's  share? 

5.  The  par  value  of  the  pound  sterling  is  $-40°-;  required 
the  value  of  £1674  at  10%  premium. 


226          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

6.  A  can    do  a  piece  of  work   in    8  days,  B  in  7  days 
and  C  in  6  days ;  in  what  time  can  they  do  it  if  all  work 
together? 

SOLUTION. — A  can  do  £,  B  -f  and  G  J  of  the  work  in  a 
day.  The  sum  of  these  fractions  is  -£$%.  If  y^  can  be 
done  in  a  day,  -J-JJ!  (the  whole)  can  be  done  in  J^=2^-|-, 
or  2  days  3^  hours. 

7.  There  are  3  pumps  placed  in  a  coffer  dam ;  one  will 
empty  it  in  10,  another  in  15  and  a  third  in  20  hours;  in 
what  time  can  it  be  emptied  by  working  all  three  at  once? 

8.  Express  ^  of  a  day  in  hours,  minutes,  etc. 

SOLUTION. — |  of  a  day  is  the  same  as  -|  of  3  days. 

Days.       Hours.       Min.       Sec. 

7)3          0  00 


10        17        8$ 

EXPLANATION. — As  7  is  not  contained  in  3  days,  we  reduce  them 
to  hours=72  hours,  which,  divided  by  7=10  hours  and  2  left,  etc. 

9.  In  £  of  a  pound  (British  money),  how  many   shil- 
lings and  pence? 

10.  In  |-  of  a  bushel,  how  many  pecks,  quarts,  etc.? 

11.  In  \  of  a  ton  (long  weight),  how  many  hundreds, 
etc.? 

12.  Find  |  of  £167  18s.  6d. 

13.  Eeduce  ^  an  inch  to  the  fraction  of  a  foot. 

14.  Reduce  -^  a  cent  to  the  fraction  of  a  dollar. 

15.  What  part  of  a  pound  Troy  is  \  an  ounce? 

16.  What  part  of  a  ton  is  J  of  a  pound? 

17.  |  of  a  farthing  is  what  part  of  a  pound? 

18.  |  of  f  of  $1600  is  what  part  of  $1000? 
Answers:  $420;  $210;  $315;  $735;    $480.00:    $3275; 

16s.  Bd.;  3  pks.  4   qts.;  3   cwt.  1   qr.  9  Ibs.  5£  oz.;  4 
hours;  $8184;  y^;  ^ ;  3^ ;  f;  3^;  -g-^. 


PROPORTION.  227 


RATIO. 

242.  The  relation  that  one  number  bears  to  another  is 
called  ratio.     The  quotient  arising  from  dividing  one  num- 
ber by  another,  of  the   same   denomination,  is   the   ratio 
between  them. 

And  as  two  quotients  can  be  obtained  from  comparing 
any  two  numbers,  it  follow*  that  two  ratios  can  also  be 
obtained.  The  relation  that  1  bears  to  2  is  J,  and  that 
which  2  bears  to  1  is  f . 

The  sign  of  ratio  is  the  colon.  The  above  ratios  would 
be  expressed  thus:  1  :  2  and  2  :  1,  and  would  be  read  one 
is  to  two  and  two  is  to  one.  Some  mathematicians  divide 
the  first  term  by  the  second;  others,  the  second  by  the 
first.  The  former  method  is  most  used: 

6  :  3  will  equal  |  or  2,     f  :  ^=i=j,  or  1|. 

i 

243.  Numbers  or  quantities  of  different  denominations 
can  not  have  a  ratio.     We  can  not  compare  3  trees  with 
5  books.     But  if  the   numbers   are  capable  of  being  re- 
duced to  the  same  denomination,  they  can  be  compared ; 
for  we  can  say  3  feet  is  to  2  inches,  as  it  is  the  same  as 
to  say,  36  inches  is  to  2  inches. 


XXVI.  PROPORTION. 

244.  Two  ratios  may  be  equal  to  each  other.     2  :  4~ 
:8. 

2  bears  the  same  relation  to  4  that  4  does  to  8. 

245.  When    ratios    are    equal,    the    numbers    or   terms 
which  compose  them  are  said  to  be  in  proportion,  and  are 
written  thus :  2  :  4  :  :  3  :  6,  and  read  2  is  to  4  as  3  is  to  6. 


228          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

The  first  and  last  terms,  as  the  2  and  6,  are  called  ex- 
tremes, and  the  second  and  third  the  means. 

246.  The  same  ratio  may  arise  by  comparing  4  quan- 
tities, two  of  which  are  different  in  denomination  from  the 
other  two. 

Tons.     Tons.  $          $ 

3     :     6     : :     6  :  12. 

The  ratio  is  J,  and  if  reversed,  as  6  :  3  :  :  12  :  6,  it 
would  be  2. 

247.  If  the  extremes  are  multiplied  together  the  pro- 
duct will  be  equal  to  the  product  of  the  means. 

3X12=36 
6x   6=36 

Hence,  when  any  3  terms  are  given,  we  can  readily  find 
the  fourth  by  dividing  the  product  by  the  odd  term.  If 
we  had  only  the  three  first  terms  of  the  above  proportion, 

Tons.    Tons.          $ 

that  is,  3  :  6  :  :  6,  the  fourth  term  would  be  found  by 
dividing  the  product  of  6X6,  or  $36,  by  3=$12,  or  thei 
fourth  term  as  above. 

To  apply  this  in  practice,  we  have  only  to  suppose  the 
3  tons  and  6  tons  to  be  coal,  and  the  $6  the  price  of  3 
tons.  Then  3  tons  is  to  6  tons,  as  the  price  of  3  tons  is 
to  the  price  of  6  tons. 

1.  What  will  35  Ibs.  of  sugar  cost,  if  7  Ibs.  cost  77  c.? 

STATEMENT. — 7  :  35  :  :  77  is  to  the  price  of  35. 

7  :  35  :  :  77  EXPLANATION. — The  means,  35  and  77, 

35  ing  multiplied    together,   produce  2G95,   and 

~^r  this  divided   by  the  given  extreme,  7,  gives 

~~.j  the  required  extreme,  385,  which  must  be  of 

.,  the   denomination  of  cents,  in   order   that   a 

7)2695  ratio  exist  between  it  and  the  third  term 

~385  cents,  ceuts- 
or  $3.85. 


PROPORTION.  229 


The  same  by  cancellation. 
5 


=385,  or  $3.85,  Am. 

By  placing  our  terms  in  fractional  form,  we  have  35X77  for  a 
numerator  and  7  for  a  denominator.  Then  reducing  both  terms  in 
the  same  ratio,  the  7  cancels  the  35,  leaving  5X77  for  a  numerator 
and  1  for  a  denominator. 

2.  If  27  £  Ibs.  of  butter  cost  $3.75,  what  will  16|  Ibs. 
cost? 

3.  Find  the  price  of  12^-  dozen  of  chickens,  at  30  cents 
a  pair. 

4.  The  price  of  21  tons,  13  cwt.,  3  qrs.  and  15  Ibs.  of 
hemp  is  $1680.55;  what  will  15  cwt.  cost? 

5.  What  will  54  Ibs.  7-J-  oz.  cost,  if  15  J  Ibs.  cost  $8.47? 

6.  If  f  of  a  ship  cost  $7000,  what  will  T9a  cost? 

7.  If  6  men   do   a  piece  of  work  in  7  days,  how  long 
will  it  take  5  men  to  do  it? 

STATEMENT.     5  :  6  :  :  7  The  7  (days)  having  no  ratio  to  the 

7  other  numbers  of  the   proportion,  is 
5VL9~  placed  to  the  right.     At  first  sight,  it 

'—  —  would  seem  that  the  proportion  would 

8  1  be  6  :  5  :  :  7;  but  6  men  do  not  bear 
the  same  ratio  to  5  men  that  the  time  of  the  6  bears  to  that  of  the 
6.     A  little  reflection  will  convince  the  learner  that  5  men  would 

I  require  a  longer  time  to  do  the  work  than  6  men,  which  would  fail 
j  to  complete  a  proportion,  as  shown  by  the  following 

j  STATEMENT.   The   greater  :  the    less  :  :  the   less  :  the   greater, 
or,  the  greater  X  the  greater=rthe  less  X  the  less! 

Hence,  to  find  the  second  term  of  a  proportion,  it  will 

jbe  necessary  to  inquire  whether  more  or  less  will  be  re- 

|  quired.     If  more,  put  the  greater  of  the  two  terms  in  the 

'second  place;  if  less,  put  the  less  of  the  two  terms  in  the 

second  place. 


230          NELSON'S  COMMON-SCHOOL  ARITHMETIC 

8.  If  two  men  plow  a  field  in  3  days,  how  long  will  it 
take  3  men  to  do  it? 

9.  If  26  yards  of  linen  cost  $13.50,  what  will  10  yards 
cost? 

10.  If  3  coats  can  be  made  from  10J   yards  of  cloth, 
how  many  can  be  made  from  31^  yards? 

11.  If  the  interest  for  $750  for  3  years,  4  months  and 
10  days  be  $151.25  (360  days  to  the  year),  what  is  it  for 
one  year? 

12.  The  interest  of  £100,  from  3d  of  April  to  25th  of 
February,  is  £6  5s.  9|^(Z. ;  what  is  it  per  year? 

13.  A,  B  and  C  are  in  partnership,  and  their  gains  for 
the  year  are  $6757 ;  what  is  each  man's  share,  suppose  A 
invested  $1567,  B  2600  and  C  3798? 

The  sum  of  their  investments  is  to  each  man's  invest- 
ment, as  the  total  gains  to  each  man's  gain. 

14.  M  invests  $6500,  N  $1487,  0  $3654;  in  3  months 
it  is  found  that  their  gains  are  $1678;  what  is  each  man's 
share  ? 

15.  A  lends  B  $1000  for  13  months  10  days;  how  long 
should  B  lend  A  $8271,  to  return  the  favor? 

16.  If  the  shadow  from  a   two-foot  rule    be    6    inches, 
what  is  the  height  of  a  tree  that  throws  a  shadow  of  75 
feet? 

17.  If  7  men  can  build  21  perches  of  masonry  in  a  day, 
how  long  will  it  take  14  men  to  build  147  perches? 

Answers:  $5.19,  9,  3£,  7,  300,  $1329.34,  $936.89,  49, 
$5.90,  $45,  $4.50,  $2.25^,  $22.50,  $7350,  $58.09,  $30.25, 
3,2. 

REMARK. — This  rule  is  of  less  utility  to  the  business  man  or  me- 
chanic than  is  generally  claimed  for  it,  as  most  of  the  problems  can 
be  solved  in  less  time,  and  with  fewer  figures,  by  the  application, 
of  Multiplication  and  Division.  Take,  for  instance,  the  17th.  If 


PROPORTION.  231 

21  perches  can  be  built  in  a  day,  147  can  be  built  in  147—7  days; 
and  if  7  men  can  do  it  in  7  days,  14  men  can  do  it  in  one-half  of 
7,  or  3J  days 

COMPOUND  PROPORTION. 

248.  A  proportion  is  said  to  be  compound  when  it  is 
composed  of  more  than  two  ratios  or  four  terms. 

1.  If  3  men  in  5  days,  by  working  8  hours  a  day,  dig 
a  cellar  15  feet  long,  12  feet  wide  and  7  feet  deep,  in  how 
many  days  will  2  men  dig  one  17  feet  long,  14  feet  wide 
and  6  feet  deep,  by  working  10  hours  a  day. 

In  this  problem  there  are  11  terms  and  5  ratios:  the 
ratio  between  men  and  men,  that  between  hours  and  hours; 
between  feet  and  feet  of  the  length ;  feet  and  feet  of  the 
width,  and  feet  and  feet  of  the  depth.  In  arranging  these 
terms,  we  proceed  as  in  simple  proportion,  Ex.  7,  page 
229. 

Days. 

Men,  2  :  3  :  :  5        1.  Days  are  wanted;  write  days  as 

Hours,  10:8  right-hand  term. 

Length  in  ft.,  15  :  17  2.  Comparing   men  with  men,  we 

Width  in  ft,  12  :  14  find  that  it  will  take  2  men  a  longer 
Deptli  in  ft.,  7  :  6  time  to  do  the  job  than  it  took  3  men, 

BO  we  write  the  greater  of  the  two  terms  (3)  in  the  second  place. 

3.  Comparing  hours  with  hours,  we  reason  that  it  will  take  less 
time  to  do  the  job  by  working  10  than  by  working  8  hours  a  day, 
BO  we  write  the  smaller  number  on  the  right  and  under  the  second 
term. 

4.  Comparing  length  with  length,  we  reason  that  it  will  take  a 
longer  time  to  dig  a  cellar  17  feet  long  than  it  did  to  dig  one  15  feefc 
long,  so  we  write  the  greater  (17)  term  under  the  second  term. 

5.  Comparing  breadth  with  breadth,  it  will  take  a  longer  time  to 
dig  a  cellar  14  feet  wide  than  it  did  to  dig  one  12  feet  wide,  so  we 
write  the  greater  (14)  under  the  second  term. 

6.  Comparing  depth  with  depth,  it  will  take  less  time  to  dig  a 
cellar  6  feet  deep  than  it  did  to  dig  one  7  feet  deep,  so  we  write  tho 
smaller  numbei  under  the  second  term. 


232          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 
The  same  by  cancellation. 


17x2 

=6$  or  6  days  8  hours.* 


$ 

$ 
17 


REMARKS. — 1.  The  pupil  should  observe  to  have  the  terms  of  each 
ratio  of  the  same  denomination. 

2.  The  answer  will  be  of  the  same  denomination  as  the  right- 
hand  term. 

2.  If  6  men   in   15  days  dig  a  trench    18  feet  long,  7 
feet  wide  and  5  feet  deep,  in  how  many  days  will  21  meu 
dig  a  trench  125  feet  long,  9  feet  wide  and  4  feet  deep? 

3.  What  is  the  interest  of  $6784  for  2  years,  6  months 
and  15  days,  at  6%  per  annum? 


STATEMENT. 


Days,       365 
Dollars,  100 


$6 
927 

6784 


Ans.  $1033.77. 

4.  The  interest  of  $1467  for  3  years,  4  months  and  12 
days  is  $450.72;  what  is  the  rate  per  cent.?f 

5.  The  interest  on  $786.55  at  10%  is  $176.44;  what  is 
the  time? 

6.  The  interest  for  a  certain  sum  of  money  for  4  years, 
2  months  and  20  days,  at  6%,  is  $100;  required  the  prin- 
cipal? 

*In  forming  this  proportion,  we  reasoned  from  what  was  given  to 
what  was  required.  For  instance,  in  comparing  men  with  men,  we 
inquired  if  it  would  take  2  men  a  longer  or  a  shorter  time  than  the 
time  (5  days)  that  it  took  3  men  to  do  it. 

fThe  pupil  can  prove  his  own  work  by  computing  the  interest  by 
the  method  taught  in  the  first  part  of  this  book. 


PARTNERSHIP*  233 


XXVII.  PARTNERSHIP. 

249.  WHEN  two  or  more  persons  associate  together  to 
carry  on  a  business,  they  are   said   to   be  in  partnership, 
and  are  called  a  firm,  house  or  company. 

Partnerships  may  be  general  or  special.  General  part- 
nerships extend  to  the  whole  of  the  mutual  dealings  of 
the  parties.  Special  partnerships  are  formed  for  some 
specific  purpose,  a  single  dealing  or  adventure. 

When  more  than  two  persons  are  engaged  in  business, 
it  is  usual  to  select  the  names  of  one  or  two  of  the  mem- 
bers, with  the  term  U0o.,"  for  the  name  of  the  partner- 
ship; as  a  business  conducted  by  Messrs.  Jones,  Evans, 
Henderson  and  Norton  might  be  called  the  firm  or  house 
of  Jones  &  Co.* 

250.  Each  member  of  a  firm  becomes  responsible  for 
the   acts  and   contracts  of  his  copartners,  in  the  way  of 
sale,  purchase,  promise,  agreement,  etc.,  performed  in  the 
course  of  the  usual  business  of  tfce  firm.     If  a  partner 
draws  a  note  or  bill  even  in  his  own  name,  and  it  can  be 
proven  to  be  on  account  of  the  partnership  business,  he 

1  thereby  renders  the  firm  liable. 

251.  An  individual  becomes  a  partner  by  allowing  the 
community  to  presume  that  he  is  such,  or  by  having  his 
name  appear  on  a  sign  or  in  a  bill,  card,  etc.     A  secret 
partner  becomes  equally  liable  when  discovered,  as  if  his 
name  appeared  in  the  firm. 

252.  A  creditor  of  one  of  the  members  of  a  firm  can 

*The  name  of  a  firm  is  not  always  derived  from  the  members 
having  the  largest  interest.  Where  precedence  is  not  given  to  age, 
the  names  of  the  most  influential  are  usually  selected. 

20 


234          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

claim  only  the  interest  of  the  debtor  in  the  partnership 
property  after  all  claims  against  it  have  been  settled. 

253.  All  partnership  agreements  should  be  written. 
The    funds,    property    and    merchandise    furnished    by 

partners  for  carrying  on  business  are  called  stock  or  cap- 
ital, and  the  gains  are  called  dividends. 

The  liabilities  of  a  partnership  or  individual  business 
are  the  debts,  and  the  assess,  their  available  means,  includ- 
ing the  indebtedness  of  others  to  them. 

An  inventory  is  a  list  or  statement  of  those  things  which 
constitute  assets. 

254.  In  keeping  partnership  accounts,  each  member  of 
the  firm  should  be  credited  with  all  that  he  brings  into 
the  concern  or  business,  and  be  charged  or  debited  with 
all  that  he  takes  out,  just  the  same  as  if  he  had  no  inter- 
est in  it. 

255.  The  calculations  peculiar  to  partnership  relate  to 
the  division  of  property  and  profits. 

1.  A,  B   and    C   have  been  in   business   one  year,  and 
find  they  have  made  a  net  gain  of  $3476,  which  is  to  be 
divided  as  follows:  A  is-  to  have  J,  B  J  and  C  J;  required 
the  share  of  each. 

$Mp._$i738j  A's  share;  8^-&=J869,  B's  share;  and 
$869=C's  share. 

2.  X,  Y  and  Z  purchase  a  tract  of  land  for  $2000,  X 
giving  $600,  Y  $900  and  Z  the  remainder.     In  one  year 
afterward  they  sell  it  for  $5500;  required   each   person's 
share  of  the  proceeds. 

3.  A,  B    and   C   invest  $2000   each.     In  three  months 
their  gross  gains  are  $2000 ;  expenses^  including  $250  for 
additional  services  of  C,  $600:   what  will  be  each  man's 
share  of  the  gain? 


V     . 


JOINT  STOCK  COMPANIES.  235 


XXVIII.  JOINT  STOCK  COMPANIES. 
STOCKS. 

296.  A  Joint  Stock  Company  is  a  body  of  men  asso- 
ciated together  in  a  species  of  partnership,  to  carry  out 
some  heavy  undertaking  requiring  the  investment  of  more 
capital  than  individuals  or  partnership  companies  com- 
monly possess.  Joint  stock  companies  are  usually  ineor-^ 
porated  by  act  of  legislature,  with  certain  privileges. 
Railroads,  canals,  bridges,  etc.,  are  generally  constructed 
by  this  species  of  combined  interest,  and  many  banking 
and  insurance  houses,  scholastic  institutions,  etc.,  are 
owned  and  managed  by  joint  stock  companies. 

When  an  association  of  this  kind  is  to  be  formed,  a  few 
leading  persons  make  an  estimate  of  the  probable  amount 
of  capital  required,  divide  it  into  equal  shares  of  from 
$10  to  $100  or  $500,  according  to  the  nature  of  the  un- 
dertaking, and  issue  certificates  of  ownership  for  each 
share.  These  are  called  certificates  of  stock,  and  are  trans- 
ferable. Persons  owning  certificates  are  called  stock- 
holders. 

Joint  stock  companies  are  usually  managed  by  a  presi- 
dent and  board  of  directors^  elected  for  the  purpose,  by  the 
stockholders. 

When  shares  sell  for  the  price  named  in  the  certificate, 
the  stock  is  said  to  be  at  par;  if  above  this  value,  they 
are  said  to  be  above  par;  if  below  it,  below  par. 

Besides  the  stocks  of  companies,  there  are  government 
stocks,  which  consist  of  bonds  that  have  been  issued  by 
state  officers,  for  the  purpose  of  borrowing  money.  These 
draw  interest  at  a  specified  rate. 


236          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

In  dividing  the  profits  of  joint  stock  companies,  it  has 
been  found  more  convenient  to  declare  the  dividend  by 
percentage. 

1.  What  is  the  cost  of  10  shares  of  railroad  stock  at 
5%  below  par,  the  original  cost  being  $100  per  share? 

Find  the  cost  of  10  shares  at  $100  and  deduct  5«£. 

2.  A   banking  institution    declares  a  dividend  of  18% 
on  a  capital  of  $30000 ;  what  amount  of  money  should  a 
stockholder  receive  who    holds    5  shares,  valued  at  $200 
each? 

3.  I  hold  15  shares  (each  $100)  of  stock  in  gas-works, 
which  have  declared  a  dividend  of  20%;  how  much  am  I 
entitled  to  after  my  gas  bill  of  $20  is  deducted? 

4.  How   many  shares  of  United   States   stocks   at    2% 
above  par   can  I  buy  for  $1224,  the   original  cost  being 
$100  per  share? 

5.  What  amount  of  stock  can  I  buy  for  $1683,  if  I  am 
allowed  2%  commission  on  the  amount  invested? 

The  amount  I  am  to  receive  is  to  be  y-g-g-  or  ^  of  the 
amount  of  stock  purchased — not  -^  of  $1683,  for  that 
would  be  commission  on  commission  and  investment. 

Let  the  amount  to  be  invested  be  represented  by  |-g, 
and  to  this  add  ^j==|^,  then  we  discover  that  $1683  is 
f-J-  of  the  amount  to  be  invested.  1 183— 33=-^,  or  my 
commission,  which,  if  we  multiply  by  50,  will  give  us  the 
amount  to  be  spent,  $1650. 

To  prove  this,  find  the  commission  on  $1650  at  2%. 

6.  A  broker  receives  1685,  which  he  is  desired  to  in- 
invest  in  State  stocks;  how  much  should  he   invest,  and 
allow  himself  2J%  on  the  investment? 

7.  What  amount  of  stock  can  a  broker  buy  for  $16700, 
and  allow  himself  J%  on  the  investment? 

Answers:  $180,  $950,  $280,  $16658,35,  $6619,51,  12. 


JOINT  STOCK  COMPANIES.  237 

257.  When  declaring  dividends,  it  is  customary  to  re- 
serve a  part  of  the  gains  of  business  for  current  expenses. 
Such  sum  is  carried  to  an  account  called  a  Contingent 
Fund  or  Contingent  Expense  Account.  Dividends,  in 
this  way,  are  usually  declared  for  an  even  rate  per  cent. 

8.  A    coal    oil    company,    with    a    capital    of   $150000, 
gain!  $31,493,  and  has  concluded  to  declare  a  dividend  of 
15%:  how  much  will  be  left  to  apply  to  contingent  fund? 

15%  of  $150000=$22500. 
$31493— $22500=$8993,  Am. 

9.  An  insurance  company  gains  $53,369.87,  in  six  mos., 
on  a  capital  of  $500000,  but  does  not  consider  it  safe  to 
declare  a  full  dividend;    how  much  will  apply  to  contin- 
gent fund  account  after  declaring  a  rate  of  10%  per  an- 
num? 

10.  What  is   the  largest  even    dividend   which    can    be 
declared  by  a  company  with  a  capital  of  $150000,  whose 
gains  are  $13547.65? 

11.  A  stockholder,  owning  20  shares,  at  $50  each,  re- 
ceives a  dividend  of  $120;  what  is  the  rate  per  cent.? 

12.  The  first  dividend  of  a  company  is  payable  in  bonds, 
by  which   a   stockholder,   owning  20    shares,   obtains   two 
shares   worth  $100  each;  what  was  the  rate  of  dividend 
declared? 

13.  What  would  apply  to  contingent  account  where  a 
dividend  of  20%   was  declared  on  a  gain  of  $316784.87, 
arid  a  capital  of  $2000000? 

Answers:  $116784.87,  10%,  12^,  9%,  $28369.87. 


238          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


XXIX.  BANKRUPTCY— INSOLVENCY. 

258.  Bankruptcy  signifies  inability  to  pay.  A  person 
becomes  bankrupt  when  he  is  obliged  to  give  up  his  busi- 
ness for  want  of  means  to  pay  his  debts,  and  to  carry  it 
on.  Such  an  individual  is  said  to  have  failed.  Bank- 
ruptcy  and  insolvency  are  synonymous  terms. 

Insolvent  debtors  usually  transfer  their  property  to 
other  parties  for  the  benefit  of  their  creditors.  This  is 
called  making  an  assignment,  and  prevents  the  individual 
debtors  from  recovering  more  than  a  share  of  the  property 
apportioned  to  the  amount  of  their  claims.  The  person 
to  whom  an  assignment  is  made  is  called  an  assignee,  the 
property  and  claims  of  the  debtor,  his  effects  or  assets,  and 
his  indebtedness,  his  liabilities. 

I.  A  person  failing  in  business  has  the  following  effects 
to  meet  claims  to  the  amount  of  $13000;  how  much 
should  his  creditors  receive  on  the  dollar?  Merchandise 
to  the  amount  of  $3500,  railroad  stock  to  the  amount  of 
$2100  and  personal  claims  to  the  amount  of  $1500. 

3500 
2100 
1500 

Amount  of  assets,  7100,  which,  reduced  to  cents,  and  di- 
vided  by   the   amount   of  the    liabilities— 5  4T8^  cents,   or 

*13|000)710|000(54T83 
65 


52_ 
~8~ 


IMPORTING.  239 

2.  The  amount  of  assets  belonging  to  an  insolvent 
debtor  is  $4684,  and  his  liabilities  $22000;  how  much  can 
he  pay  on  the  dollar? 


XXX.  IMPORTING. 

259.  IMPORTING  is  the  business  of  buying  goods  in  a 
foreign  to  sell  in  a  home  market.     A  tax,  under  the  name 
of  duties  or  customs,  is  imposed  by  government  on  most 
imported  articles  of  commerce.     Such  taxes  are  levied  for 
the  purpose  of  creating  revenue  to  defray  the  expenses  of 
government  or  to  protect  home  manufactures  and  agricul- 
tural interests.     Duties  are  regulated  by  a  scale  of  prices 
called  a  tariff]  and  are  altered  according  to  the  exigencies 
of  the  times  or  caprice  of  the  administration. 

A  liigli  tariff  signifies  high  rates  of  duties,  and  a  low 
tariff,  low  rates  of  duties.* 

The  persons  appointed  to  examine  imported  goods  and 
collect  taxes  are  called  custom-house  officers,  and  their  place 
of  business,  the  custom-house. 

260.  Duties  are  of  two  kinds:  Ad  valorem  and  Specific. 
Ad  valorem  duties  consist  of  a  rate  per  cent,  on  the  value 
of  goods  as  stated    in   the  invoice;    Specific  duties,  of  a 
stated  sum  of  money  on  the  quantity  imported,  without  re- 
gard to  value,  as  $1  a  gallon,  $20  a  ton. 

261.  Certain  allowances,  called  draft,  tare,  leakage  and 
breakage,  are  made  on  goods  charged  with  specific  duties. 
These  allowances  sometimes  consist  of  a  percentage  of  the 
weight  or  quantity  and  sometimes  of  a  specific  deduction. 

Tare  is  an  allowance  made  for  the  weight  of  the  box, 

*  Taxes  are  often  levied  upon  exports  as  well  as  imports. 


240          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

barrel,  bag,  crate,  etc.,  which  contains  the  goods,  and  is 
usually  calculated  by  percentage;  etc.,  after  the  deduction 
for  draft  is  made. 

Draft  or  tret  is  an  allowance  made  for  loss  by  weighing 
in  small  quantities,  and  for  impurities  to  which  some 
goods  are  subject. 

On       112  Ibs.,  or  less,  it  is  1  Ib. 

From  112    "     to    224  Ibs.,  2  Ibs. 

224    "      "     336    "     3    " 

336    "      "  1120    "     4    " 

1120    "      "  2016    "     7    " 

More  than      2016     "     9     " 

Leakage  is  an  allowance  of  2%  on  liquids  in  casks, 
paying  duties  by  the  gallon. 

Breakage  is  an  allowance  on  bottled  liquors,  usually 
5%,  but  on  ale,  beer  and  porter,  10%. 

Gross  Weight  is  the  total  weight  of  goods  and  box,  bar- 
rel, etc. 

Net  Weight  is  what  remains  after  all  deductions  are 
made. 

We  shall  not  trouble  the  learner  to  work  out  any  ques- 
tions in  this  chapter,  as  it  rarely  happens  that  young  peo- 
ple have  them  to  do  in  business. 


XXXI.  FOREIGN  EXCHANGE. 

262.  IN  calculating  Foreign  Exchange,  the  money  of 
one  country  has  to  be  represented  in  that  of  another.  A 
bill  drawn  in  New  York  on  a  merchant  in  England  will 
be  expressed  in  pounds,  shillings  and  pence. 

The  draft,  though  not  stated  in  the  question,  is  to  be  deducted  bo- 
fore  other  allowances  are  made. 


FOREIGN  EXCHANGE.  241 

263.  The  relative  value  of  moneys  of  different  coun- 
tries  depends  on   the  par  of  exchange  and   the  course  of 
exchange. 

264.  The  par  of  exchange  is  the  comparative  value  of 
the  coins  of  the  different  countries,  and  is  fixed,  while  the 
relative  purity  of  the  coins  is  the  same.     The  par  of  ex- 
change between   the  United   States   and  Great  Britain   is 
$4.86  to  the  pound  sterling.     Formerly,  when  the  silver 
of  the   United   States    was   purer,    the   par   value   of  the 
pound  sterling  was  4|,  and  exchange  is  still  quoted  from 
this  par,  or  9^  premium  on  the  old  currency  being  the 
present  par  value. 

265.  The  course  of  exchange  usually  depends  upon  the 
relative  state  of  indebtedness  of  the  merchants  of  the  dif- 
ferent countries,  and  the  supply  of  gold  and  silver;    ac- 
cordingly, the  course  of  exchange  will  sometime  be  above 
and  sometimes  below  par. 

EXCHANGE  WITH  GREAT  BRITAIN. 

FORM  OP  A  FOREIGN   BILL. 

Exchange  for  £1567.  CINCINNATI,  June  3,  1867. 

Thirty  days  after  sight  of  this  first  of  Exchange,  (sec- 
ond and  third  of  the  same  tenor  and  date  unpaid,)  pay  to 
the  order  of  William  Tuechter,  the  sum  of  One  thousand 
five  hundred  and  sixty-seven  pounds  sterling,  value  re- 
ceived, and  place  to  my  account  as  advised.* 

To  William  Morgan,  JSsq.,  J.   B.   TREVOR. 

Liverpool,  England. 

*  Foreign  bills  are  generally  drawn  in  sets  of  two,  three  or  four; 
that  is,  copies  of  the  same  bill  are  made  out  and  transmitted  by 
different  conveyances  to  the  payer,  one  of  which  being  received  and 
accepted,  or  paid,  the  others  to  be  void.  These  copies  are  called 
First,  Second  or  Third  of  Exchange.  The  above  is  a  copy  of  the  first. 
The  others  are  drawn  in  a  similar  manner. 


242 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


BRITISH   OR  STERLING  MONEY   REDUCED  TO 
FEDERAL   MONEY. 

GIVING  the  cost  of  British  money  is  called  quoting  it, 
and  the  rate,  the  quotation. 

Sometimes  the  total  cost  of  a  pound  is  given;  at  other 
times,  only  the  premium  on  the  old  par. 

266.    When  the  Amount  is  given  in  the  Quotation. 

1.  What  is  the  value  of  £157  9  2  in  Federal  money  @ 
$4.86  to  the  pound  sterling? 

SOLUTION. — Since  £1  is  equal  to  $4.86,  £157  will  be 
equal  to  157  times  486,  or  $763.02.  Then  taking  aliquot 
parts  of  a  pound,  we  have  6  shillings  and  8  pence:=J, 
and  2  shillings  and  6  pence^^J  of  a  £;  add  J  and  -J  of 
$4.86  to  $763.02=765.25. 

TABLE  OF  ALIQUOT  PARTS 


Of  a  Pound. 

G>/  a  Pound. 

0/  a  Shilling 

s.      d. 

d. 

d. 

10     0  is   i 

10     is    7V 

6    is  £ 

6     8 

8 

i 
sir 

4 

4 

5     0 

I 

7J 

3 

i 

4     0 

6 

5 

2 

3    4 

£ 

5 

A 

I* 

^ 

2     6 

i 

4 

1 

T 

2     0 

3 

A 

Of 

1     8 

_C 

2 

0^ 

^4 

1     4 

IT 

1J 

yi(5 

0£  < 

1     3 

A 

1 

T¥0 

1     0 

7>Tf 

2  to  4.  At  $4.84,  what  will  the  following  sums  amount 
to?  £345  14  0,  £15  3  4,  £365  7  6. 

5  to  7.  £147  13  4  @  $4.87,  £425  17  6  @  $4.44,  £652 
10  0  @  $5=?  Answers:  $5872.53,  $3515.02 

267.  2b  reduce  British  to  Federal  money  when  the  Pre* 
mium  only  is  given. 


FOREIGN  EXCHANGE.  243 

8.  What  is  the  value  of  £221  15  6  @  8%  premium? 
The  premium  is  reckoned  on  the  old  par  value;  that  is, 
$4.44|  to  the  pound,  or  $40  to  £9. 

12)6.0         Reducing  15  shillings  and  6  pence  to  the  decimal  of  a 
Poun<*j  we  have  -775. 


775 

Hence,  £221  15  6=r-£221.775,  which,  multiplied  by  40 
and  divided  by  9=$985.666,  to  which,  if  we  add  8%,  we 
shall  have  for  the  answer,  $1064.52. 

What  is  the  value  of  the  following  in  Federal  money? 

9.  £1424  19  9  @  7J%       13.     £313     8  4  @  S%% 

10.  £3575  18  6  @  8  %       14.     £505  19  6  @  9  % 

11.  £1100  12  6  @  8J%       15.  £3737  12  3 

12.  £111  @  9f  #       16.     £649     4  6 

Total,  $29793.55.  Total,  $25270.28. 

267.  When  gold  is  at  a  premium,  the  rate  may  be 
added  after  the  rate  of  exchange. 

Take  the  8th  example,  assuming  gold  to  be  at  a  pre- 
mium of  45  cents,  or  45%. 

Value  of  £221  15  6  at  the  old  par^$985.666. 

To  this  add  8%  premium  on  British  money  and  45% 
on  gold  to  the  sum,  and  we  have  $1543.56. 

268.  TO   REDUCE    FEDERAL    TO    BRITISH    OR 
STERLING  MONEY. 

CASE  I. 

When  the  Amount  is  given. 

17.  At  $4.87  to  the  pound  sterling,  what  will  be  the 
value  of  $37654? 

Since  $4.87=£1,  $37654  will  be  the  equal  to  as  many 
pounds  as  $4.87  is  contained  times  in  that  number. 
W65400-r-487==7731.827f  which  reduced=£7731   16  6J. 


244          KELSON'S  COMMON-SCHOOL  ARITHMETIC. 

Reduce  the  following  to  British  money: 

18.  $3674.87  @  $4.87.  19.  $67845.18  @  4.44|. 

Total,  £16019  15  2. 

CASE  II. 

WJien  the  Premium  only  is  given. 

20.  At  9%  premium  on  sterling  money,  what  will  be 
the  value  of  $3964? 

At  9%  premium  $1.09  is  worth  only  $1;  therefore, 
$3964  are  worth  as  many  dollars  as  $1.09  is  contained 
times  in  it. 

396400^-109=3636.697,  which,  reduced  to  pounds  by 
multiplying  by  9  and  dividing  by  40=818.2568,  or 
£818  5  1J. 

21  and  22.  Reduce  the  following  to  British  money : 
$3165  @  §\%,  $1678.90  @  9£%.  Total,  £996  2  2J, 


XXXII.  DTSITRAETCE. 

269.  INSURANCE  is  a  guarantee  against  loss.     It  may 
be  of  several  kinds,  as  Fire,  Marine  and  Life  insurance. 

270.  Insurance  on  fixed  property  is  called  fire  insur- 
ance;   that   on  movable   property,   as  goods  in   course  of 
transportation,  ships,  etc.,  is  called  marine  insurance;  that 
which  guarantees  the  payment  of  a   sum   of  money   to  a 
survivor  at  the  death  of  an  individual,  life  insurance. 

271.  The  act  of  insuring  is  termed  taking  a  risk;  the 
amount  paid    for   insuring,   the  premium;    arid  the  paper 
upon  which  the  contract  is  written,  the  policy. 

272.  When  the  risk  is  heavy,  the   insurer  sometimes 
re-insures  in  another  company. 

273.  In  time   of  war,  the  rates   of  insurance  increase 


INSURANCE.  245 

with  the  danger  to  which  the  property  is  exposed,  or  else 
the  company  secures  itself  by  inserting  in  the  policy  ex- 
ceptional matter  called  the  war  clause. 

The  rates  of  insurance  vary  according  to  the  exposure 
of  the  property  and  the  character  of  the  property  itself; 
the  greater  the  risk,  the  higher  the  rate. 

Insurance  can  be  obtained  from  one  day  to  a  term  of 
years,  giving  a  range  of  rates,  from  a  small  fraction  of  one 
per  cent,  to  three,  four  and  even  higher  rates  per  cent. 

1.  How  much  should  be  paid  to  insure  a  house  valued 
at  $1674,  premium  being  IT»%,  and  policy  $1.50? 

2.  At    2^%    premium,   what   should    I    pay    on    $6710 
worth  of  goods? 

3.  At  4%    premium,  what  should  I  pay  on  machinery 
and  material  in  a  factory,  the  estimated  value  of  which  is 
$6600? 

4.  A   company   takes   a   risk   of  $35000  in  a  block  of 
buildings,  at  l-g-%,  and  re-insures  $15000  in  another  com- 
pany at  l|r%;  how  much  premium  does  it  realize? 

5.  At  y1^  of  \%    for  10  days,  what  should   I   pay  on 
$30000? 

6.  I  have  insured  $16000  for  3  months  at  T5a  of  \%\ 
how  much  should  I  pay? 

7.  What  will  be  the  insurance  on   merchandise   worth 
£675,  to  be  shipped   from  Liverpool  to  New  York,  at  4: 
guineas  per  cent.? 

Answers:  £28  7s.,  $26.61,  $167.75,  £30.5,  $80,  $30, 
$264.00,  $337.50. 


246         NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


XXXIII.  DUODECIMALS. 

274.  DUODECIMALS,  like  decimals,  is  a  species  of  cal- 
culation which  enables  the  operator  to  compute  fractional 
quantities  as  whole  numbers. 

12""  fourths  make  1  third. 
12'"    thirds    make  1  second. 
12"    seconds  make  1  prime,  or  inch. 
12'     primes  or  inches  make  1  foot. 
1  inch     is  the  fa  of  a  foot. 

1  second  is  the  j1^  of  an  inch,  or   y^     of  a  foot. 
1  third    is  the  fa  of  a  second,      jfa-g    of  a  foot. 
1  fourth  is  the  fa  of  a  third,  or  2^-^g-  of  a  foot. 
As   applied   to   mechanical   pursuits,    duodecimals   have 
seldom  to  be  subtracted  or  divided;  hence,  the  following 
exercises  will   be  confined  to  multiplication  and  addition 
exclusively. 

1.  Multiply  2  ft.  5  in.  by  3  ft.  4  in 

25  !•  Writing  the  dimensions  as  in  the  margin,  we  com- 

3     4  mence  with  the  left-hand  figure  (3  feet),  and  say,  3 

^ — jr times  5  inches  are  15  inches,  or  1  foot  3  inches;  write 

Q     Q        3  in  the  inches'  place. 

2.  Then  3  times  2  are  6,  and  the  1  foot  carried  makes 

808        7  feet,  which  we  write  in  the  place  for  feet. 

3.  We  next  multiply  by  the  4  inches;  that  is,  we  multiply  5  inches 
or  y5£  of  a  foot  by  4  inches,  or  -4^  of  a  foot.     The  result  will  be  -f-fr, 
but  to  avoid  fractions,  we  call  the  result  20"  or  1  inch  8".     Write 
8  inches  to  the  right  of  and  below  the  3  inches. 

4.  Then  4  times  2  are  8',  and  !'=$'  or  inches,  which  we  write  in 
the  inches'  place. 

6.  We  now  proceed  to  add  them.  There  being  nothing  to  add  to 
8",  we  set  it  down ;  then  9'  and  3'  are  12'  or  inches=l  foot.  Write 
0  and  add  1  to  the  7,  makes  8  feet:  the  product  of  2  ft.  5  in.X3  ft.  4 
in.=:8  0'  8'. 


DUODECIMALS.  247 

ANOTHER  WAY. 

2.  Multiply  the  following   dimensions   together  :  10  ft. 
7  in.X3  ft.  8  in.x7  ft.  9  in. 

10         .7  Here  we  commence  to  multiply  by 

3          8  the  left-hand  figure  (3),  and  write  the 

~~3fi        9?  result  without  reducing  to  a  higher 

denomination.      3X10  ft=30  ft.,  and 
7  in.X3=21  in.     Then  multiplying  by 


30      101        56  1st  pro.      the  8,  we  write  the  first  product  un- 
7          9  der  itself  as  the  multiplier,  and  the 


210      707      392  second  product,  56,  one  place  further 

270      909     504          ^°  ^ie  r*&kt.     Adding  these,  we  have 

the  product  of  two  divisors. 

Proceeding  in  the  same  way  with 

the  7  and  9  of  the  third  dimension,  we  add  together  the  products 
and  reduce  them  to  higher  denominations,  by  which  we  get  300  ft. 
8'  11",  or  300 J*2  ft.+Jj1¥=300J  ft.,  nearly  * 

ft.  in.        ft.     in.  ft.     in.     ft.  in.      ft.  in. 

3.  17  IX  3    4=?  7.     4     8X6  4x17  2=? 

4.  14  6X  7     8=?  8.     3     9X2  6x11  0=? 

5.  21  9X14  11-=?  9.  21  11X6  7x17  8=? 

6.  18  8X16     7= 

Total  answers :  802  1'  3"  and  3159  6'  3"  8'". 

ft.  in.       ft.     in.  •  ft.    in.     ft.  in.    ft.  in. 

10.  21  7XH  10=-?  13.  31  7X3  2x3  3==? 

11.  13  9x17     4=?  14.  26  3x9  5x7  7=? 

12.  33  7X29    3=?  15.  17  1x6  7x0  9=? 
Total  answers:  1476  7",  2283  10'  9"  6'". 

Mechanics  preferring  common  or  decimal  fractions  to 
duodecimals,  seldom  use  the  latter. 

The  following  example  is  worked  by  both  methods: 

*For  this  very  simple  method,  we  are  indebted  to  J,  C.  Kinney, 
Esq.,  of  Reading,  Ohio,  not  having  seen  it  before.  Its  simplicity 
would  suggest  it  as  the  best  method  to  teach  this  otherwise  difficult 
rule. 


248          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

16.  How  many  squares  of  flooring  in  3  rooms  measuring 
18  ft.  G  in.Xby  15  ft.  8  in.,  and  what  is  the  cost  of  lay- 
ing, at  50  cents  per  square? 

¥-X^Xi=869£  ft.,  or  8.695  sqs. 

18     6 
15     8 


277     6 
12     4 


289  10 
3 

869  6,  or  869J  sq.  ft.,  which,  reduced  to 
squares  of  100  feetr=8.695  squares.  8.695x50  cents— 
4.347,  or  $4.35. 

17.  What  is  the  cost  of  laying  4  floors  of  the  following 
dimensions,  at  75  cents  per  square?     18  ft.  9  in.Xl7  ft.  3  in. 

18.  What  will   be   the  cost  of  shingling  a  roof  which 
measures  53  ft.  6  in.  long,  and  5  ft.  8.  in.  from  the  ridge 
to  the  outer  edge  of  the  wall,  at  $1.50  per  square? 

19.  The  average  breadth  of  a  board  is  1  ft.  4  in.,  and 
the  length  23  ft.  9  in.;  what  number  of  feet  does  it  con- 
tain? 

20.  How  many  solid  feet  in  a  log  measuring  as  follows? 
45  ft.  4  in.Xl  ft.  6  in.Xl  ft-  3  in. 

Answers:  $9.09,  85  ft.,  $9.70,  31|  sq.  ft. 


XXXIV.  INVOLUTION— EVOLUTION. 

275.  THE  process  of  multiplying  a  number  by  itself  a 
certain  number  of  times  is  called  Involution,  while  that  of 
finding  the  number  thus  raised,  or  the  reverse  process,  is 
called  Evolution. 


IN  VOLUTION— EVOLUTION.  949 

276.  A  number  multiplied  upon  itself  is  raised  to  the 
power;  the  second  power  multiplied  by  the  number 

is  raised  to  the  third  power;  the  number  of  the  power  be- 
ing indicated  by  the  number  of  times  the  original  number 
has  been  used. 

The  second  power  is  also  called  the  square,  because  the 
number  of  square  feet,  inches,  etc.,  is  found  by  multiply- 
ing the  number  contained  in  one  side  by  itself.  For  a 
similar  reason  the  third  power  is  called  the  cube. 

277.  The  power  of  a  number  is  indicated  by  a  small 
figure  over  the  right  of  the  number,  thus :  53,  which  shows 
that  the   third  power   of  5  is  understood.     This  figure  is 
called  the  index  or  exponent. 

278.  Decimals  are  raised  to  any  power  in  the  same  way 
as  whole  numbers,  with  the  difference  of  placing  the  deci- 
mal point,  while  common  fractions  are  involved  by  multi- 
plying the  numerators  and  denominators  separately.     The 
second  power  of  .5  is  .5X-5  or  -25,  and  the  second  power 

offis|x|=?V 

279-  The  product  of  two  numbers  can  not  consist  of 
more  figures  than  there  are  in  the  two  factors,  and  can 
consist  of  only  one  less  than  in  the  two  factors.  Take  9,  the 
largest  number  of  a  single  figure,  and  multiplied  by  itself, 
it  produces  only  two  figures,  81 ;  and  take  10,  the  small- 
est number  of  two  figures,  and  multiply  it  by  itself,  and 
it  produces  three  figures.  This  principle  applies  in  find- 
ing the  roots  of  numbers. 

1.  Square  the  following  numbers:  3,  7,  9,  4,  6,  15,  27, 
89,  97,  112. 

2.  Raise  the  following  numbers  to  the  powers  indicated : 
3s  95,  26*,  305,  87*,  2503,  1893. 

280.  The  number  from  which  any  power  is  raised  is 
called -the  root  of  that  power,  and  the  process  of  finding 
that  number  is  called  extracting  the  root. 


250          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

281.  The  root  of  a  number  derives  its  name  from  the 
exponent  of  the    power,  the  second  or  square  root  being 
from  the  second  power,  the  third  or   cube  root  from    the 
third  power;   and  is  indicated  thus:  -j/,  square  root;  ^K, 
cube  or  third  root;  j^,  the  fourth  root. 

282.  A  complete  power  is  one  which  can  have  its  root 
extracted.     A  surd  is   one  which  can  not   have   its   root 
extracted.     4  is  a  complete  power,  while  5  is  a  surd. 

TABLE. 

THE   SQUARE   ROOT   OP 

1=1  36=  6  121=11  256=16 

4=2  49=  7  144=12  289=17 

9=3  64=  8  169=13  324=18 

16=4  81=  9  196=14  361=19 

25=5  100=10  225=15  400=20 

283.  Since   the   product  of  any  two  numbers   can  not 
consist  of  more  than  four  or  less  than  three  digits  or  fig- 
ures, the   number   of  which   a  root  is   composed  can   be 
found  by  separating  the  squares  into  periods  of  two  figures 
each,  thus :  256  indicates  that  two  figures  formed  the  root, 
2,56  being  the  periods,  the  root  of  which  is  16;  and  in 
the  same  way  746496  is  pointed,  74,64,96,  indicating  that 
three  figures  composed  the  root.     The  square  root  is  864. 

THE  EXTRACTION  OF  THE  SQUARE  ROOT. 

1.  The  square  root  of  765625  is  how  much? 

76,56,25(875 

64  1.  Commencing  at  the  right-hand,  the  power  is 

1£7    M  9"R          separated  into  periods  of  two. 

7     1 TfiQ  ^*  ^e  nearest  S(luare  root  of  the  last  period  is 

then  taken,  which  gives  8,   or  800.       [Art.  83.] 

1745     )8725      Writing  the  8  in  the  quotient,  and  squaring  the 
8725     number,  we  have  64  (6400),  which  is  written  un- 
der the  76,  or  76,00,00 


INVOLUTION— EVOLUTION. 


251 


3.  Subtracting  this  64  from  the  76,  we  have  a  remainder  of  12,  to 
which  another  period  (56)  is  annexed,  making  1256. 

4.  For  a  part  of  the  new  divisor,  the  8  of  the  quotient  is  doubled, 
giving  16  as  a  trial  divisor.     Finding  it  is  contained  7  times,  (the 
7  being  included  in  the  divisor,)  that  figure  is  annexed,  making 
167,  and  the  product  is  completed. 

5.  1169  being  subtracted  from  1256,  leaves  87. 

6.  Annexing  the  last  two  figures  of  the  dividend,  the  last  figure 
of  the  divisor  is  doubled,  as  before,   making   174.     This  number, 
with  the  last  figure  of  the  quotient  (5)  annexed,  is  contained  in 
8725,  5  times  without  a  remainder,  making  the  square  root  875. 

RECAPITULATION. — Separating  the  power  into  periods,  we  find  the 
highest  root  of  the  last  period.  This  we  place  as  the  first  quotient 
figure,  and  subtract  its  square  from  the  period.  To  the  remainder 
annex  the  next  period,  and  for  a  trial  divisor  double  the  last  figure 
of  the  divisor.  To  this  divisor  we  annex  the  next  quotient  figure 
and  multiply  as  in  long  division.  To  the  next  remainder  is  an- 
nexed the  next  period,  and  to  the  last  divisor  is  added  its  last 
figure,  which  is  the  same  as  to  double  the  quotient,  and  the  opera- 
tion proceeds  as  before. 

2.  -t/1683129  is  how  much? 


1,68,31,29(1297.354 


22 

2 


249 
9 


2587 
7 

25943~ 
3 


259465 
5 


2594704 


68 
44 


2431 
2241 


19029 
18109 


92000 

77829 


1417100 
1297325 


NOTE  1. — This  answer  may  be 
carried  out  to  any  number  of  places 
by  annexing  ciphers,  as  has  been 
done  to  produce  the  .354  of  the  quo- 
tient. Three  figures,  however,  are 
sufficiently  correct  for  practical 
purposes. 

2.  To  find  the  square  root  of  a 
decimal  quantity,  we  commence  at 
the  left  to  point  off  the  periods  of 
two  figures.  146.739  would  be 
pointed  thus:  1,46.73,90. 


11977500 
10378816 


252          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

The  pupil  can  prove  the  accuracy  of  his  calculations  by 
squaring  the  root  obtained. 

3.  v/14161.  8.  j/16820.17. 


4.  1/625.  9.  1/23467.809. 

5.  !/ 99980001.  10.  ^167037^827 


6.  j/99999.8000001.  11.  T/456789.375. 

7.  y/7837619.  12.  |/10963.849. 

284.  The  square  root  of  a  fractional  number  is  found 
by  extracting  the  root  of  each  term.     The  square  root  of 

sV:=V/247= 1- 

285.  Decimals  are  pointed  off  in  periods  from  the  right. 
.31671  is  pointed  thus:  .31,67,10. 

286.  The  square  root  of  the  product  of  two  numbers 
gives    a    mean    proportional    between    them.     |/5x20= 
1/100—10,  the  mean  proportional  between  5  and  20. 

287.  The  square  root  of  the  area  of  a  square  is  equal 
to  the  length  of  the  side.* 

13  to  17.  Find   the  mean  proportional   between   7  and 
175,  121  and  36,  6  and  24,  42  and  38,  16  and  49. 
Answers:  35,  28,  66,  39.949+,  12. 

18.  A  square  garden  contains  2916  yards;  what  is  the 
length  of  a  side  in  feet? 

19.  A   pavement   is    112   feet  long  and  7  feet  broad; 
what  will  be  the  length  of  the  side  of  a  square  of  equal 
area? 

20.  How  many  yards  of  ground  in  the  side  of  a  square 
which  would  be  equal  to  a  lot  measuring  144x196? 

21.  What  is  the  length  of  the  side  of  a  square  piece  of 
land  which  contains  25600  acres? 

Answers:  28,  168,  162,  160,  2023.85+. 

*The  area  is  the  contents  of  the  surface,  or  the  number  obtained 
by  squaring  the  side  of  a  square. 


INVOLUTION—  EVOLUTION.  253 

288*  The  surfaces  of  circles  are  to  each  other  as  the 
squares  of  their  diameters  or  circumferences. 

289.  A    triangle    is    a    figure   having 
three  sides  and  three  angles,  or  corners. 
If  one  of  these    angles    is   square,   it  is 
called    right   angle    and    the    triangle    is 
called  a  right-angled  triangle. 

290.  Two  square  figures,  one  having 
each  of  its  sides  equal  to  the  perpendic- 
ular, and   the   other  having   each  of   its 

sides  equal  to  the  base,  will,  together,  be  equal  to  a  square, 
each  of  whose  sides  is  the  same  length  of  the  hypotheuuse.* 
Let  the  perpendicular  be  3,  the  base  4  and  the  hypoth- 
enuse  5. 

Then  3X3=  9,  the  area  of  the  first  square. 
4X4=16,     "       "     "     "    second  " 
25,     "       "     "     "    third     " 

But  25  is  the  hypothenuse  squared  or  multiplied  by  it- 
self; therefore  the  square  root  of  the  square  25  will  be  the 
length  of  the  hypothenuse.  The  square  root  of  25=5. 

Hence,  the  square  root  of  the  sum  of  the  squares  of  the 
base  and  perpendicular  will  give  the  hypothenuse. 

And  the  square  root  of  the  difference  ,of  the  squares  of 
the  hypothenuse  and  either  of  the  two  sides  will  give  the 
third  side. 

22.  What  length  of  a  ladder  will  reach  across  a  15-foot 
alley  to  the  top  of  a  house  30  feet  high? 


1125,  the  square  root  of  which  is  33  ft.  6J  t?i.,  nearly. 

*The  pupil  should  construct  a  diagram,  with  these  squares 
his  slate. 


254          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

23.  What  is  the  diagonal  of  a  room  18  feet  by  16? 

24.  A   ladder   30    ft.  long,    placed    between    two   trees, 
reaches  to  the  height  of  27  feet  on  one  of  them  and  25 
on  the  other;  what  is  the  distance  between  them? 


XXXV.  EXTRACTION  OF  THE  CUBE  ROOT. 

291.  THE  cube  root  of  a  number  is  such  a  number 
which,  if  multiplied  upon  its  square,  will  make  that  num- 
ber :  22X2,  or  23r=8.  The  cube  root  of  8=2.  The  sign 
of  the  cube  root  is  -j*K. 

The  cube  root  of  any  number  consisting  of  three  figures 
will  be  a  number  represented  by  one  figure  ;  the  cube  root 
of  a  number  containing  more  than  three  and  less  than 
seven  figures,  will  be  one  containing  two  figures  ;  hence, 
we  point  off  the  figures  by  threes  instead  of  twos,  as  iu 
square  root.  N 

1.  Find  the  cube  root  of  262144. 

262    144(64        EXPLANATION.  —  The  near- 
g3  —  216  est  cube  root  of  the  first  be- 


—10800    '  placed  to  the  left  and  its 

6        x30-     79?  cube  taken'   which  is  216 

P  X*A^  (216000).     This    subtracted 

.  _  from    the    dividend,   leaves 

11536  46144  46,  to  which  is  annexed  the 

next  period   (144),  making 
46144. 

For  a  trial  divisor,  63  is  then  multiplied  by  300,  giving  10800, 
which  is  contained  in  46144,  4  times. 

The  part  of  the  root  previously  obtained  is  multiplied  by  this, 
and  that  product  by  30,  giving  720.  The  square  of  the  last,  figure 
of  the  root  is  then  taken,  and  the  three  results  added,  making  11536, 
which,  multiplied  by  the  last  figure  in  the  quotient,  gives  46144. 


EXTRACTION  OF  CUBE  ROOT.  255 

2.  Find  the  cube  root  of  17596287801. 

17V596V2875801(2601 
2*=   8 

22X300     =1200  )9596 
2  X<>X30=  360 
62  =    36 

1596    9576 


20287801 

2602X300  =20280000 
260X1X30=  7800 
I2  = _1 

20287801        20287801 

This  differs  from  the  last  example  only  in  the  cipher  of  the  quo- 
tient. 

Finding  the  trial  divisor  (262X300)  was  not  contained  in  the  new 
dividend,  a  cipher  was  annexed  to  the  quotient,  and  another  to  the 
trial  divisor,  giving  2602X300  or  20280000. 

This  being  contained  in  the  dividend  1  time,  the  former  part  of  the 
quotient  was  multiplied  by  it  and  30,  and  with  the  square  of  the 
last  figure  (1)  added  to  the  trial  divisor,  as  before,  giving  20287801, 
which,  multiplied  by  1,  completed  the  extraction.  The  cube  root 
is  2601. 

RULE. 

Find  the  greatest  root  of  the  left  period,  place  it  in  the  quo- 
tient and  divisor ,  and  subtract  its  cube  from  the  dividend. 

To  the  remainder  annex  the  next  period,  and,  for  a  trial 
divisor,  multiply  the  square  of  the  root  thus  obtained  by  300. 

Divide  the  new  dividend  by  this  divisor,  and  enter  the 
product  of  it,  with  the  root  already  obtained  and  30,  under 
the  divisor ;  under  this  enter  the  square  of  the  last  quotient 
fyure,  and  the  sum  of  the  three  numbers  will  be  the  true 
divisor. 

Multiply  this  divisor  by  the  last  quotient  fyure,  and  s?/7>- 
tract  the  product  from  the  dividend;  to  the  remainder  annex 
another  period,  and  proceed  as  before. 


25G          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

3  to  7.  -^389017=?  ^259696072=?  ^5735339=? 
^219365327791=?  f  99252847=  ? 

APPLICATION  OF  CUBE  ROOT. 

292.  A  cube  is  a  solid  body,  having  all  its  sides  of 
equal  length.  Any  two  sides  of  a  cube  multiplied  to- 
gether will  give  the  superficial  contents  of  one  of  the  faces 
of  the  cube,  and  this  multiplied  by  another  side,  will  give 
the  solid  contents;  therefore,  the  cube  root  of  the  number 
of  feet,  yards,  etc..  contained  iu  any  solid,  will  give  the 
side  of  a  cube  of  equal  bulk. 

7.  An  irregular  block  of  stone    contains    15781    cubic 
feet  and   1333   cubic  inches,  or  27270901    cubic   inches; 
what  will  be  the  side  of  a  cube  of  equal  solidity? 

8.  Required  the  depth  of  a  cubic  cistern  that  will  con- 
tain 3375  feet. 

9.  What  will  be  the  side  of  a  cubic  bin  or  box  that 
will  contain  20  bushels  of  wheat? 

Answers:  15  ft.;  35.03  in.;  301  in.  or  25  ft.  1  in 


XXXVI.   AEITHMETIC    APPLIED    TO    THE 
TRADES,  FARMING,  ETC. 

CARPENTRY. 

THE  Carpenter  proper  may  be  called  the  outside  and  the 
joiner  the  inside  carpenter.  The  distinction,  is  seldom  ob- 
served. 

Master  Carpenters  are  sometimes  called  Builders.  They 
will  contract  for  the  entire  work  of  an  edifice,  and  super- 
intend its  construction.  The  legitimate  business  of  the 
carpenter  is  to  prepare  and  fit  all  the  wood-work  used  in 


CARPENTRY.  257 

building  houses.     His  prices  depend  on  the  quality  of  ma- 
terial and  style  of  finish. 

Plain  work  on  one  side  of  white  pine  lumber  is  taken  as  the 

unit  of  measurement,  and  is  called 1 

Plain  work  on  poplar  is 1J 

Plain  work  on  ash,  oak,  etc 2.} 

Plain  work  on  maple 3 

Segmentat  or  Norman  work  on  white  pine 2J 

Gothic  work  on  white  pine 3 

Serpentine,  or  the  Oriental  variety,  plain 5 

Domes 9 

Floors,  roofs,  partitions  and  weatherboarding  are  meas- 
ured and  charged  for  by  the  square — 100  sq.  ft. 

The  quantity  of  good  lumber  required  for  a  square  of 
flooring  is  112  feet,  -J-  or  12 J%  being  allowed  for  waste. 

The  quantity  of  good  pine  shingles  required  for  a  square 
of  roofing  is  1000. 

USEFUL  HINTS  ON  BUILDING. 

Persons  about  to  erect  buildings  in  cities,  should  obtain  a  permit 
from  the  Board  of  Improvement,  else  they  subject  themselves  to 
damages  for  placing  obstructions  in  the  street. 

Proprietors  of  adjoining  lots  and  buildings  should  be  notified  of 
the  digging  of  cellars,  or  the  making  of  other  excavations  that 
would  endanger  their  property.  In  Cincinnati,  cellars  may  be  dug 
12  feet  deep  without,  the  risk  of  incurring  damage.  To  excavate  or 
build  cellars  deeper  than  this,  or  to  build  sub-cellars,  permission 
should  be  obtained  from  adjoining  proprietors. 

Where  property  is  valuable,  it  will  be  to  the  advantage  of  persons 
who  purpose  building,  to  remember  that  vaults,  cellars,  cisterns,  etc., 
may  be  said  to  occupy  no  room;  and  that  license  may  be  obtained 
from  the  Board  of  Improvement  to  extend  cellars  or  vaults  under 
the  sidewalk  or  pavement. 

Foundation  walls  should  be  made  thicker  than  the  main  walls  of 
buildings,  and  the  latter,  to  be  secure,  should  rest  on  the  middle 
of  the  foundation,  allowing  it  to  project  on  both  sides. 


258 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


PRICES  OF  CARPENTER  WORK, 

WITH  C10LU  AT  PAR,  FOR  A  DWELLING-HOUSE  OF  TEN  ROOMS. 


Cellar  windows,  usual  size, 

with  sash $1.50 

Cellar  steps,  good 2.00 

"      doors,  each ,...  4.00 

First  floor  joists,  IJin.,  per 

square,, , 0.75 

Floors,  per  square 0.65 

Trimmers,  per  foot 0.10 

pinning,       "      "  0.02 

Roofing,  per  square 1.50 

Hip  rafte*,  per  foot 0.06 

Valley  "         "      "   0.06 

Cants „ 0.01 

"     0.03 

Trap-doors,  each 1.50 

Ceiling  joists,  per  square...  0.50 
Partitions,  «  "  ...  0.35 
Poor  heads,  «  "  ...  0.35 
Inside  door-frames,  per  ft..  0,04 
Outside  "  "  "..  0.05 

Window  frames,  plain 0.04 

Beads  for  do.,  soft  wood....  0.01 
"        "     '*     hard     "    ....  0.01  J 

Box.  W.  frames,  soft 0.08 

"     »         «        hard 0.11 

«     "        "        beads 0.01  J 

R.  sills, ,.. 0.06 

M.  rails..,.! 0.06 

Betting  frames .,,,,.,.,.,.  0.07 

Pocket  and  pullies,  perpr..  0.35 
Hanging  sashes,  per  pair..  0.20 
Base,  6  in.  wide,  plain,  per 

lineal  foot ,..,.. 0,05 

JJase  mold..,..,.... 0.08 

<<         "   large. „ ,.  0.09 

Casings,  6  in 0.05 

"        5  »  ..,., 0.04 

Plinths,  each., , 0.10 

Caps,    7  in.,  per  foot ,0.10 

"       8  "       '<      "    0.08 

"     10  "       «     «   M 0.10 

«     Cornice 0.25 

^helves,  conimon,.,.,. ,  0.03 


Shelves,  best $0.05 

Cloak  rails. 0.03 

Trimming  doors 0.30 

Door  sills 0.20 

Mantels,  each 5.00 

"         common,  each...<(  4.00 
Cupboard  front,  per  sq.  ft..  0.04 

Window  stools,  per  foot 0.06 

"  "         «      «   0.18 

"  «     large 0.22 

Cornices. 
Gutters  on  eave  inverted, 

per  lineal  foot 0.13 

Do.,  mold 0.18 

Do.,  3  members 0.23 

Do.,  4         "         0.25 

Do.,  mold 0.37 

Do.,     "    brackets 0.40 

Do.,  6  members 0.30 

Do,,  6  modillions,  etc 0.42 

Do.,  truss,  each $3  to  5.00 

Truss  usually  referred  to 
measurer. 

Fastening  ornaments 0.04 

Bracket  cornices 0.05 

Lining  from $0,04  to  0.05 

Tubes 0.25 

Porticoes. 
Square  colums,  8  to  9  in., 

per  foot 0.15 

Capitals,  each 0.75 

Best 1.25 

Porch,  front,  per  foot 0.15 

"        panel 0.20 

"        extra 0.35 

'<       cornice,  plain 0.30 

"        full..,,,, 0.50 

Ornamental  left  to  meas- 
urer. 
Floors  and  roofs,  per  sq.  ft.  0.25 

Framework 0.20 

Sills O.tt 


MASONS'  WORK.  259 

In  carpenter  work,  the  cost  of  material  is  about  equal 
to  the  cost  of  labor.  For  calculations,  see  duodecimals, 
page  246. 

MASONS'  WORK. 

293.  A  bill  of  prices  and  a  standard  of  measurement 
are  generally  fixed  upon  by  mechanics  of  the  various  cities 
of  the  Union,  and  contract  work  is  charged  for  at  a  cer- 
tain rate  per  cent,  on  the  bill,  according  to  agreement. 

Master  masons,  carpenters,  etc.,  usually  select  one  of 
their  number  who  is  expert  in  figures,  and  a  good  judge 
of  work,  to  attend  to  the  measurement  of  work  done. 
This  person  is  called  a  measurer,  and  his  fee  is  paid  one- 
half  by  the  workmen  and  the  other  by  the  employer.  The 
following  rules  are  taken  from  the  Cincinnati  Stone-ma- 
sons' bill  of  prices: 

"RULES  FOR  MEASURING. 

"1.  All  work  is  to  be  measured  by  the  perch  of  24J 
cubic  feet. 

"  2.  All  work  to  be  measured  from  each  outside  corner, 
including  all  openings  under  eight  feet  wide. 

"3.  All  openings  less  than  five  feet  wide  to  measure 
solid,  and  round  their  jambs,  provided  there  is  no  frame; 
but  if  there  is  a  frame,  they  measure  solid  and  half  round 
their  jambs. 

"4.  Chimney  abutments  and  common  pillars  to  meas- 
ure front  and  both  ends  for  length. 

"5.  All  walls,  however  thin,  to  be  reckoned  eighteen 
inches  thick. 

"  6.  All  partition  walls  to  be  measured  from  out  to 
out."* 

*  "NOTE. — The  above  rules  are  for  workmanship  only.     When  the 


260          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

When  measuring  hewn  stone,  neatly  piled  before  it  is 
laid  in  the  wall,  and  in  computing  masonry  for  public 
works,  it  is  customary  to  reckon  25  feet  to  the  perch. 
Twenty  per  cent,  is  deducted  for  loose  stone  in  piles. 

1.  How  many  perches  in  a  pile  of  hewn  stone  meas- 
uring 28  feet  long,  5  feet  high  and  4  feet  broad? 

28X5X4 

-  ---  =22^<>,  or  22  per.  10  ft. 


2.  What  quantity  of  stone   is  in    a   pile   150x15x12 
feet? 

3.  What  will  it  cost  to  build  the  foundation  of  a  house, 
which  is  75  feet  long,  16  feet  wide  and  7  feet  high,  wall 
18  inches  thick,  @  $2.25  a.  perch,  including  materials? 

4.  What  will   it  cost  to  build   a  cellar  that  is  16  feet 
square,  with  walls  2  fe,et  thick  and  8  feet  high,  @  82.75 
per  perch,  material  included? 

Answers:  $173.73.  1080  perches,  $113.77. 

BRICKLAYERS'  WORK. 

294.  Bricklayers'  work  is  computed  by  the  thousand 
bricks.  The  usual  dimensions  of  a  brick  are  8  inches 
long,  4  inches  broad  and  2  inches  thick.  There  are  21 
bricks  in  a  cubic  foot  of  wall,  mortar  included.  The  gov- 
ernment standard  is  1000  bricks  to  40  cubic  feet.  A 
brick  of  the  above  dimensions  weighs  4^  Ibs. 

A  bushel  of  sand  weighs  113  pouuds  ;  of  slacked  lime, 
51  pounds. 

workman  furnishes  materials,  the  value  of  the  materials,  both  stone 
and  mortar,  in  the  above  extra  measurements,  as  well  as  the  value 
of  the  materials  saved  by  doors  and  windows,  is  to  be  deducted 
from  the  foot  of  the  bill;  that  is,  neither  stone  nor  mortar  is  to  be 
charged  in  any  instance  where  they  are  not  used." 


BRICKLAYERS'  WORK.  261 


RULES    OF    MEASUREMENT    OF    THE    BRICK- 
LAYERS OF  CINCINNATI. 

"All  lengths  shall  be  exterior  or  taken  on  the  outside, 
from  corner  to  corner,  and  for  every  return  or  cross  sec- 
tion at  openings  deducted,  nine  inches  by  height  for  work 
and  materials,  or  one  foot  by  height  for  workmanship 
shall  be  allowed.  Autaes  and  Pilasters  returns  allowed 
in  all  cases. 

"  All  octagon  or  circular  work  of  a  radius  of  three  feefc 
or  less  shall  measure  double,  and  for  larger  radius  a  less 
but  fair  allowance  shall  be  made.  All  walls  cut  up  for  or 
coped  with  brick,  shall  measure  one  foot  additional  height, 
and  all  walls  both  cut  up  for  and  coped  with  brick,  two 
feet  additional  height.  All  work  with  a  batter  or  mate- 
rial deviation  from  plumb  line,  shall  measure  once  and  a 
half.  • 

"Flemish  or  plumb  bond  fronts  shall  measure  solid  in 
all  cases,  and  stock  or  pressed  brick  with  tuck  joints  shall 
measure  double,  with  the  additional  cost  of  pressed  brick 
if  furnished  by  the  contractor. 

"  Kettles  or  stills  shall  measure  solid,  and  those  of  50 
feet  or  less,  exterior  surface,  once  and  a  half;  and  those 
of  25  feet  or  less,  exterior  surface,  double.  Fire  fronts, 
when  cased,  double  measurement. 

"Culverts  or  sewers,  9  inches  thick  and  two  feet  or  less 
i  in  diameter,  and  those  of  half-brick  thick  and  three  feet 
or  less  in  diameter,  shall  measure  solid.  All  circular 
work  shall  measure  exterior  girth. 

"All  openings  of  sectional  area  greater  than  ten  feet 
;ii  shall  be  deducted,  except  those  in  bond  fronts  and  ovens, 
which  shall  not  till  exceeding  fifty  feet;  then  half  of  such 
;  |  shall  be  deducted. 


362          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

"Twenty-one  bricks  shall  be  allowed  to  the  cubic  foot 
for  all  brick-work,  and  the  same  proportion  for  thicker 
or  thinner  walls  from  one  brick  in  thickness  upward; 
forty- three  bricks  shall  be  allowed  per  yard  for  brick 
paving  when  flat,  and  eighty  for  paving  on  edge  in  ascer- 
taining the  amount  of  bricks  only;  but  when  paving  is 
done  and  measured  as  brick-work,  the  sand  shall  be  al- 
lowed for  in  the  measurement,  and  all  materials  on  edge 
or  cut  to,  waste  allowed  for  also. 

"Cisterns,  when  measured  for  brick-work  and  materials, 
half  brick  wall  shall  count  six  inches  and  whole  brick  wall 
ten  inches,  and  cistern  arches  and  those  over  circular 
vaults  shall  in  all  cases  measure  double. 

"The  floor  joists  shall  govern  the  height  of  stories  in 
all  cases  till  two  stories  make  more  than  twenty-four  feet 
for  carrying  materials,  and  then  twenty-four  feet  shall  be 
allowed  for  two  stories  and  each  ten  feet  of  the  additional 
height**  a  story.  When  new  work  is  built  upon  an  old 
building,  or  one  built  by  another  contractor,  all  such  work 
shall  be  measured  and  allowed  for  an  additional  story  in 
height  for  labor  of  carrying  materials. 

"Old  bricks  in  piles  shall  be  subject  to  the  usual  dis- 
count in  ascertaining  the  quantity,  and  in  the  absence  of 
special  agreement  shall,  if  on  the  premises  and  sound,  be 
valued  same  as  new  bricks  at  the  kiln;  if  from  a  burnt 
building,  or  otherwise  unsound',  shall  be  valued  by  the 
measurer  accordingly. 

"When  hands  or  materials  are  furnished  for  an  em- 
ployer, in  the  absence  of  agreement  or  contract,  actual 
cost,  together  with  fifteen  per  cent,  thereon,  shall  be 
charged  by  the  contractor. 

"Lumber  and  materials  for  scaffolding,  mortar  beds,  etc., 
and  vessels  for  holding  water,  shall,  in  all  cases,  be  fur- 
nished by  the  employer,  unless  otherwise  agreed  upon,  iu 


STONE-CUTTERS.  263 

•which  case  a  reasonable  charge  shall  be  made  by  the  con* 
tractor. 

"For  all  work  not  embraced »in,  or  provided  for  by  the 
foregoing,  a  fair  and  reasonable  allowance  shall  be  made 
by  the  measurer." 

1.  Plow  many  thousands   of  brick  will   be   required  to 
build  a  wall  90  feet  long,  6  feet  high  and  20  inches  thick? 

90X6=540  square  feet  of  surface. 

In  one  square  foot  of  a  20-inch  wall  there  are  35  bricks  j 
in  540  square  feet  there  are  540x35  =  18900  bricks,  which, 
divided  by  1000=18.9,  or  18T90. 

2.  In   a  house   there   are   6200  square  feet  of  20-inch 
wall  and  2000  square  feet  of  12-inch  wall;  what  will  be 
the  cost  of  building,  at  11.50  a  thousand,  including  price 
of  brick  and  laying? 

3.  What  will  it  cost  to  pave  a  yard  25  by  50  feet,  and 
a  walk  75  by  5  feet,  @  50  cents  a  yard   including  mate- 
rials ? 

Answers:  $90.28,  2978.50. 

4.  A  cistern  is  8  feet  in  diameter  and  12  feet  deep  (av- 
erage measure) ;  what  will  be  the  cost  of  building  at  40 
cents  a  barrel? 

82X12X.1865=143.23  bbls.,  143.23x40c.=$57.29. 

Instead  of  multiplying  by  .7854  (as  required  by  Art. 
),  and  dividing  by  4211  (the  number  of  feet  in  a  bar- 
rel), we  merely  use  the  quotient  arising  from  .7854-7- 
4.211:=r.l865,  as  a  multiplier. 

The  true  pitch  of  a  roof  is  obtained  by  making  the  rafters  three- 
fourths  of  the  width  of  the  building. 

The  Gothic  pitch  is  that  produced  by  making  the  rafters  as  long 
as  the  building  is  wide. 

For  information  relating  to  the  department  of  Building,  we  are 
indebted  to  R.  B.  Moore,  Esq. 


264          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


STONE  CUTTERS. 

AVERAGE    PRICE    AND    RULES    OF  MEASUREMENT  FOR   COM- 
MON FREESTONE   WORK. 

295.  In  measuring  plain  stone  work,  all  the  dressed 
faces  of  the  stone  are  taken,  and  the  whole  reduced  to  su- 
perficial measurement.-  For  instance,  a  step  4  feet  long, 
14  inches  wide  and  7  inches  thick  would  be  measured  as 
follows : 

Length,    4  ft.-f  7  in. -(-7  in.=r:5  ft.  2  in. 
Width,  14      +7  =1       9, 

and  5  ft.  2  in.Xl  ft.  9  in.=9  ft.  1",  or  9  ft. 

REMARK. — It  will  be  observed  that  the  ends  have  been  measured 
twice;  this  is  in  accordance  with  custom. 

Window-sills  are  measured  by  the  running  foot,  includ- 
ing the  projections  of  the  ends.  Prices,  for  7-inch  wide, 
and  4  to  5  thick,  per  foot,  18  to  25  cents. 

Water-table. — The  stone  in  front  of  a  house  and  on  a  level 
with  the  door-step.  Measured  as  above,  37-^  cents  per  foot. 

Ashler  or  slab  front. — Face  measure,  adding  all  worked 
ends.  Price,  40  cents  per  superficial  foot.* 

Flagging. — Superficial  measure,  2^-  inches  thick,  25 
cents;  3  inches,  30  cents;  4  inches,  40  cents;  6  inches,  50 
cents;  8  inches,  60  cents. 

Fire-wall  doping,  running  measure,  11  by  2  inches,  per 
foot,  25  cents. 

Chimney  coping,  2^  to  3  inches  thick,  per  foot,  30  cents. 

Coping  caps,  common  size,  each,  $2. 

Hearths,  common  thickness,  per  superficial  foot,  40  cents. 

*  Stone-cutters  set  (build)  their  own  work,  the  charge  for  which 
is  included  in  these  prices. 


PLASTERING.  265 

Edge  curlings,  for  walks,  2  to  3  inches  thick,  per  linear 
foot,  25  cents. 

Door  and  window  cornice,  not  exceeding  6  inches  thick 
and  6  inches  projection,  per  foot,  37-J-  cents.  Measured 
length  and  returns  3  girts  from  wall  to  wall. 

Piers  for  open  fronts,  face  measure,  taking  the  girt. 

A  common  door-piece,  comprising  2  each,  plinths,  piers 
and  caps,  with  lintel,  cornice  and  blocking,  will  cost  from 
$50  to  $75. 

All  cut  stone  should  be  laid  in  cement. 

Foundations  and  excavations  for  steps  should  be  sunk 
at  least  three  feet  below  the  surface,  else  the  action  of  the 
frost  on  the  earth  will  be  liable  to  displace  the  stone. 

Mortar  should  not  be  exposed  to  the  action  of  frost 
until  it  is  set. 

During  the  heats  of  summer,  mortar  is  injured  by  a  too 
rapid  drying  in  the  wall;  to  prevent  this,  the  other  ma- 
terials, stone  or  brick,  should  be  thoroughly  moistened 
before  being  laid;  and  afterward,  if  the  weather  is  very 
hot,  the  masonry  should  be  kept  wet  until  the  mortar 
gives  indications  of  setting.  In  very  warm  weather,  the 
top  course  should  always  be  well  moistened  by  the  work- 
men on  quitting  their  work  for  any  short  period. 

PLASTERING. 

296.  The  business  of  the  plasterer  is  to  cover  brick 
and  stone  work,  ceilings  and  partitions,  with  plaster,  and 
prepare  them  for  paper,  paint,  etc. ;  also,  to  form  cornices 
and  such  other  decorative  portions  of  walls  and  ceilings  as 
may  be  executed  in  plaster  or  cement. 

To  lay  off  a  square  corner  or  a  right  angle,  with  a  carpenter's  rule: 
Measure  3  feet  from  the  corner  in  one  direction,  4  feet  in  another 
direction  and  separate  the  points  5  feet  apart. 

23 


266          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


"CINCINNATI  PLASTERERS'   RULES   OF  MEAS- 
UREMENT. 

"  1.  All  work  shall  be  measured  superficially,  including 
openings.  All  heights  shall  be  taken  from  the  floor  to  the 
ceiling. 

U2.  All  staircases  eight  feet  wide  and  under  shall  be 
measured  double;  all  over  eight  feet,  once  and  a  half. 

"3.  All  passages  four  feet  wide  and  under  shall  be 
measured  once  and  a  half;  all  over  four  feet,  once  and  a 
fourth. 

"4.  All  inclined  ceilings  to  measure  once  and  a  half. 

"5.  All  dormor  windows,  closets  and  privies  to  be  meas- 
ured double. 

"  6.  All  octagon  and  circular  work,  except  ceilings  of 
rooms,  to  be  measured  double.  All  arched  ceilings  of 
rooms  to  be  measured  once  and  a  half. 

"7.  The  deductions  for  openings  occasioned  by  doors 
and  windows,  when  the  workman  furnishes  materials,  shall 
be,  for  lathwork,  one-eighth;  for  brick  walls,  one-fourth. 

"8.  The  materials  for  scaffolding  and  mortar-beds  and 
vessels  for  holding  water,  are,  in  all  cases,  to  be  furnished 
by  the  employer." 

STUCCO  WORK. 

Moldings,  cornices  not  over  12  inches  girth,  are  meas- 
ured by  the  running  or  linear  foot.  Flowers^  sometimes 
singly;  in  moldings,  per  superficial  foot. 

Eighteen  laths  will  cover  a  yard;  500  laths,  a  square 
of  100  feet. 

Estimate  work  is  made  from  a  bill  of  prices,  the  carpenter  agree- 
ing to  work  for  a  certain  percentage  on  the  bill. 


PAINTING,  PAPER-HANGING  AND  GLAZING         267 

1.  "What  will  be  the  cost  of  plastering  a  room  18  feet 
long  and  16  feet  wide,  with  a  ceiling  10  feet  high,  at  18 
cents? 

18 
J.6 

34x^—68,  length  round  the  room. 
68x10=680  square  feet  in  walls. 
18X16=288       "         "     "   ceiling. 

9)968 
107.55  yds.  at  18c.=$19.359,  or  $19.36. 

2.  How  many  square   yards  of  plastering  in  a  hall  84 
feet  long  and  40   feet  wide,  with  a  ceiling  18  feet  high, 
having  a, space  of  600  square  feet  occupied  by  windows? 

Ans.  802f  yards. 

PAINTING,  PAPER-HANGING  AND  GLAZING. 

297.  Painters'  Work  is  measured  by  the  square  yard, 
and  charged  for  according  to  the   number   of  coats,  the 
quality   of  the    paint    and    the    description    of  the   work. 
Sash  frames  are  charged  for  singly  or  by  the  piece,  and 
sashes  by  the  number  of  squares.     Lettering  is   charged 

.for  by  the  lineal  foot.  Common  lettering,  25  cents;  gild- 
ing, 75  cents. 

Painting  is  sometimes  charged  for  by  the  quantity  of 
paint  used,  and  the  time  spent  in  putting  it  on. 

The  calculations  being  so  simple,  it  is  considered  un- 
necessary to  give  any  examples. 

298.  Glazing  is  sometimes  charged  by  the  square  foot, 
and  sometimes  so   much   per  light.     When  estimated   by 
the  foot,  it  is  usual  to  include  the  sash  in  the  measure- 
ment. 

299.  Paper-hanging  is  charged  for  by  the  piece.     The 
commoner  qualities  measure  about  7f  yards  by  19  inches, 


268          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

or  about  35  square  feet;  the  better  qualities,  say  from  50 
cents  a  piece,  are  9  yards  by  21  inches,  or  47  square  feet. 
Border  paper  is  made  in  rolls  of  the  same  dimensions  as 
the  wall  paper,  each  roll  or  piece  containing  two  or  more 
strips,  each  of  which  is  called  a  piece,  and  is  sold  at  about 
the  same  price  as  the  paper  it  is  designed  to  match. 

Dealers  in  paper  usually  contract  for  the  hanging,  and 
charge  from  20  to  25  cents  a  piece,  according  to  quality. 
300.  To  find  the  quantity  of  paper  required  for  a 
room,  compute  the  number  of  square  feet  in  the  walls; 
deduct  the  openings  and  divide  the  result  by  the  number 
of  feet  in  a  piece.  The  space  occupied  by  the  base  will 
allow  for  waste  and  matching  pattern. 

1.  A  room  is  16  feet  square  and  has  a  ceiling  12  feet 
high,  with  two  doors  7  feet  by  4,  two  windows  7  by  3 
feet  and  a  fire-place  4  feet  square;  how  much  paper  will 
be  required  to  hang  it,  and  what  will  be  the  whole  cost, 
including  hanging — paper  25,  border  35  cents? 

16  X   4=64,  length  of  wall  around  the  room. 

64X12=768  square  feet  of  wall 
Windows,    42  feet. 
Doors,          56     "  @ 

Fire-place,  16    " 

114 

654  square  feet  to  be  covered. 

654 

=19  pieces. 

35       number  of  feet  in  a  piece. 

19X25  cents=cost  of  paper 4.75 

Length  of  border,  64  feet,  or  3  pieces  at  35 1.05 

Hanging  22  pieces  @  20  cents,* 4.40 

Total  cost $10.20 

*  The  border  is  included  in  the  22  pieces. 


FARMING.  269 


GAS  FITTING  AND  PLUMBING. 

301.  Gas  fitting  is  charged  for  per  foot  of  pipe,  vary- 
ing according  to  size.     Fittings  and  chandeliers,  per  piece. 

302.  Plumbing  is    charged    for  like    gas    fitting.     For 
ordinary  house  work,  say  from  25  to  30  cents  per  foot  of 
pipe ;  sheeting  by  weight. 

FARMING. 

The  young  farmer  will  find  it  to  his  interest  to  be  a 
good  arithmetician.  For  those  who  have  not  had  the  ad- 
vantages of  an  early  education,  we  will  introduce  a  few  of 
the  simpler  and  more  necessary  calculations,  suggesting, 
at  the  same  time,  that  during  his  leisure  moments  the 
farmer  should  master  the  entire  science  as  contained  in 
this  little  work,  which  any  person  of  ordinary  ability,  who 
can  read  and  write,  may  accomplish  without  the  aid  of  a 
teacher. 

303.  To  find  the  number  of  acres  in  a  field   or  tract 
of  land    having    four    square    corners,*   we    multiply    the 
length  by  the  breadth,  and  divide   the  result  by   160,  if 
the  measure  was  taken  in  rods;  or  by  43560,  if  taken  in 
feet.f 

1.  The  length  of  a  field  is  125  rods  and  its  breadth  112 
rods;  how  many  acres  are  in  it? 

112X1^5=14000,  which,  divided  by  160,  gives  87J. 

*  A  figure  having  square  corners,  and  all  its  sides  equal,  is  a 
equare;  one  having  its  opposite  sides  equal,  a  rectangle  or  parallelo- 
gram. 

tin  a  square  rod  there  are  272|-  square  feet.  When  there  are  feet 
remaining  to  be  reduced  to  rods,  it  will  be  sufficiently  accurate  to 
divide  by  272. 


270          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

2.  A  lot  of  land  is  400  feet  long  by  110  feet  broad; 
how  many  acres  does  it  contain?          Ans.  1  acre  1.7  rods. 

304.  To  lay  off  a  given  quantity  of  land. 

3.  What  should  be  the  length  of  a  strip  of  land  30  rods 
broad  to  contain  6  acres? 

In  6  acres  there  are  960  rods,  which,  divided  by  30= 
32  rods. 

305.  To  find  the  contents  of  a  field  in  the  shape  of  ct 
right-angled  triangle,  we  multiply  the  two  shorter  sides  to- 
gether, and  take  one-half  the  product. 

REASON. — A  right-angled  triangle  is  half  a  square  or  parallelo- 
gram, formed  by  drawing  a  line  between  opposite  corners. 

4.  The  shorter  sides  of  a  right-angled  triangle  are  45 
and  60;  required  the  contents.  Am.  1350. 

306.  To  find  the  quantity  of  grain  or  coal  in  a  bin  or 
wagon,   we   multiply   the    length,  breadth   and    height  to- 
gether, and  for  grain  divide   the  product   by  1.2444,*  if 
the  diminsions  are  given  in  feet;  or  by  2150.42, f  if  given 
in  inches.     For  coal,  by  1.555,  or  2688. 

5.  A  wagon  is  8  feet  long,  5  feet  broad  and  18  inches 
deep;  how  many  bushels  of  corn  does  it  contain? 

8X^X1^=60,  the  number  of  cubic  feet. 
1.2444)60.0000(48.21,  or  48£  bushels.* 
49776 

102240  NOTE. — Two  ciphers  were  annexed  to  the 

99552  dividend  to  correspond  with  the  decimals  of 
the  divisor,  and  produce  the  ivhole  numbers,  48, 
and  two  more  ciphers  were  annexed  to  pro- 
duce the  decimal  .23,  or  -275L  or  i. 

19920 

6.  How  many  bushels   of  grain   in   a  bin  measuring  4 
feet  every  way?  Ans.  51^,  nearly. 

*Feet  in  a  bushel.  t  Inches  in  a  bushel. 


FARMING.  271 

307.  To  find  flie  quantity  of  wood  or  bark  in  a  pile,  we 
multiply  the  three  sides  given  in  feet,  as  before,  and  di- 
vide by  128,  the  number  of  feet  in  a  cord. 

7.  How  many  cords  of  wood  in  a  pile  40  feet  long,  7 
feet  high  and  4  feet  broad?  Ans.  8|-  cords. 

SOS.  Having  two  sides  and  the  contents  of  a  box,  to  find 
the  third  side,  we  divide  the  cubical  contents  by  the  pro- 
duct of  the  two  sides. 

REASON. — Since  the  product  of  the  three  sides  equals  the  con- 
tents, the  contents  divided  by  two  of  the  sides  will  give  the  third 
side. 

8.  A  box  is  2  feet  wide  and   3  feet  high;   how  long 
should  it  be  to  hold  25  bushels  of  coal  ? 

In  25  bushels  there  are  2688X25  or  67200  cubic  inches. 
In  2  ft.  there  are  24  inches. 

"  3  "      «       "    36      "      24x36=864=area  of  the  end. 
672-5-864==77|5|,  or  6  ft.  5f  in. 

9.  What  must  be  the  height  of  a  bin  that  will  hold  300 
bushels  of  wheat,  if  its-  length  is  30  feet  and  its  width  4 
feet?  Ans.  3  ft.  1J  in. 

10.  What  must  be  the  depth  of  a  box  16  inches  square 
to  hold  a  bushel?  a  box  10  inches  square  to  hold  a  peck? 
one  8  inches  square  to  hold  half  a  peck? 

To  find  the  side  of  a  cube  that  will  hold  a  certain  quan- 
tity. See  Cube  Root. 

309.  To  find  the  quantity  of  grain  when  heaped  against 
a  wall  or  partition,  take  half  the  perpendicular  height  for 
one  side,  and  multiply  it  by  the  length  and  breadth,  as  in 
Art.  306. 

310.  To  find  the  number  of  cubic  feet  in  a  round  log. 
See  Art.  on  the  Cylinder.     To  find  the  solidity  of  a  cyl- 
inder, wa  multiply  the  area  of  the  end  by  the  length. 


272          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

11.  How  many  feet  are  in  a  log  12  feet  long   and  30 
inches  in  diameter? 

In  30  inches  there  are  2J,  or  2.5  feet:  2.5X2.5X-7S54 
=4.9087,  the  area  of  the  end.  4.9087X12=58.9044,  or 
58T9Q  feet,  the  solid  contents. 

REMARK. — This  method  of  calculating,  though  correct,  is  seldom 
used  for  practical  purposes.  It  is  customary  for  lumber  merchants 
to  throw  off  one-third  of  the  diameter,  and  consider  the  remainder 
the  side  of  a  square  log.  A  log  of  the  dimensions  named  in  the 
preceding  question  would  thus  measure  only  33J  feet,  or  one-third 
of  100  feet;  and  is  thereby  taken  as  the  standard  of  measurement 
in  some  of  the  Western  States.  See  Lumber  Business. 

311.  Trade,  or  barter. 

12.  How  many  cords  of  wood,  at  S3. 75  a  cord,  should 
I  get  for  50  bushels  of  wheat  at  $1.12J  a  bushel? 

50x1-12^=^56.25,  which,  divided  by  $3.75,  will  give 
the  number  of  cords.     5625-i-375— 15  cords. 
PROOF.— 15  cords  at  $3.75=856.25. 

13.  How  many  pounds   of  sugar,  at  8  cents  a  pound, 
should   I   get  for  127  pounds  of  buttor,  at   12J   cents  a 
pound?  Ans.  198£. 

14.  How  many  days'  work  of  a  man,  at  75  cents  a  day, 
will  be  equal  to  45  days'  work  of  a  man  at  $1.25? 

Ans.  75. 

15.  How  many  cords  of  wood,  at  $2.25,  will  be  equal 
to  150  cords,  at  $3.50? 

16.  How  many  yards  of  muslin,  at  8  cents  a  yard,  can 
be  bought  for  5  dozen  chickens,  at  $1.25,  and  15  dozen 
eggs,  at  8J-  cents? 

LUMBER  BUSINESS. 

312.  Lumber  measure  comprises    solid    and   superficial 
measure.     Round  logs  are  measured  by  deducting  one-third 


LUMBER  MEASURE.  273 

of  the  diameter  for  waste,  and  calling  the  remainder  the 
side  of  a  square  log. 

1.  To  find  the  contents  of  a  round  log  24  inches  in  di* 
ameter  and  30  feet  in  length. 

SOLUTION.— Deducting  J  from  24  for  waste,  we  have  16, 
which,  squared=256  inches,  and  multiplied  by  the  length^ 
640  feet  board  measure. 

In  some  places  only  J  is  deducted  for  pine  lumber.* 

Planks  or  joists  are  sometimes  reckoned  by  face  meas- 
ure; that  is,  the  dimensions  of  one  side  of  the  board  are 
taken  instead  of  the  solid  contents.  A  16-foot  board  2 
inches  thick  by  12  inches  broad  would  measure  32  feet 
board  measure,  or  16  feet  face  measure. 

In  some  places,  the  saw-log  is  taken  as  a  standard  of 
measurement  for  round  timber.  A  log  12  feet  long  and 
30  inches  in  diameter  is  the  standard  in  some  parts  of  the 
west.  In  Pennsylvania,  a  saw-log  is  one  that  will  cut  into 
200  feet  of  lumber. 

313.   To  measure  timber  partly  squared,  it  is  customary 

to  deduct  the  "wane"  (the  length  of  the  corner)  from  the 

thickness  of  the  log,  and  call  the  remainder  one  side.     A 

•log  18   inches  thick,  with  a  "wane"  3  inches,  would  be 

called  one  of  18  by  15  inches. 

2.  In  an  octagonal   log,  25   feet  long  20  inches  thick, 
with  a  wane  4  inches,  how  many  solid  feet  are  there? 

Ans.  55|. 

3.  There  are  150  logs,  the  average  length  and  breadth 
of  which  are  20  feet  by  22  inches,  wane  3  inches ;  required 
the  number  of  solid  feet  they  contain.  Ans.  8708J. 

*  Inch  measure  is  taken  as  the  standard  for  lumber.  If  a  board 
is  under  an  inch,  it  is  measured  as  a  full  inch;  and  if  over  an  inch, 
it  is  reduced  to  inch  measurement.  A  plank  2  inches  thick  would 
be  considered  as  two  boards  1  inch  thick. 


274  NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

4.  In  a  raft  there  are  450  boards  16  feet  long  and  1^ 
inches   thick,   and   measuring   in   the  aggregate   757    feet 
broad;  how  many  feet  of  lumber  (board  measure)  does  it 
contain?     How  many  face  measure? 

Ans.  18168  board  measure,  12112  face  measure. 

5.  How  much  lumber  can  be  cut  from  a  tree  measuring 
20  feet  long  and  14  inches  diameter  at  the  smaller  end, 
allowing  for  waste  one-fourth  of  the  diameter? 

HOUSEKEEPING. 

314.  Housekeepers,  and  ladies  generally,  ought  to  be 
familiar  with  the  operations  in  arithmetic  which  apply  in 
computing  house  rent,  servants'  wages,  board  bills,  inter- 
est, the  quantity  of  carpet  to  cover  a  floor  or  paper  for  a 
room,  etc. 

HOUSE  RENT. 

Landlords,  in  renting  by  the  year,  usually  collect  their 
rents  quarterly  ;  but  when  renting  monthly,  collect  monthly. 

By  a  quarter  is  meant  three  calendar  months.  As,  for 
instance,  if  a  house  is  rented  on  the  17th  of  April,  the 
quarter  would  expire  on  the  17th  of  July. 

When  a  house  is  rented  for  a  year,  the  tenant  is  liable 
for  the  rent  during  the  whole  of  that  time  unless  the 
landlord  accepts  another  in  his  stead.  A  verbal  lease  for 
a  year  is  binding.  A  lease  for  three  or  more  years  should 
be  recorded. 

Tenancy  begins  on  obtaining  possession.  When  there 
is  a  lease,  however,  and  the  time  not  stated,  it  is  pre- 
sumed to  commence  on  the  date  of  the  instrument. 

When  the  tenant  does  not  remove  at  the  end  of  the 
year,  or  two  weeks  afterward,  he  will  be  regarded  as  hav- 
ing rented  for  another  year. 

Interest  can  be  collected  on  rent  from  the  day  it  is  due. 


HOUSEKEEPING.  275 

A  tenant  is  released  when  the  landlord  accepts  a  sub- 
stitute. 

A  married"  woman  can  not  make  a  lease  or  take  one  in 
her  own  name. 

A  tenant  at  will  is  liable  for  rent  as  long  as  he  occu- 
pies the  premises. 

For  interest  calculations,  see  page  143. 

1.  A  house  which  rents  at  $75  a   month    is   occupied 
from  January  3  to  February  9 ;  what   is   the   amount  of 
rent? 

Rent  for  1  month=  75  , 

"       "    6dsor$=  15 

$90 

2.  Required  the  rent  for  a  house  from  April  3  to  Au- 
gust 5,  at  $1000  a  year. 

3  mo=  1  1000  EXPLANATION.— From  April  8  to  July  3 

~7^:T is  3  months,  or  1  fourth  of  a  year;  1  fourth 

of  $1000  gives  $250.     From  July  3  to  Au- 
gust 5  is  1  month  and  2  days;   30  days  is 
1  third  of  3  months,  and  the  rent  for  that 
338.888    time  is  $83.333,  and  2  days  is  1  fifteenth 
or  $338.89     of  a  month,  giving  the  rent  for  that  time, 
$5.555. 

3.  Required  the  rent  for  a  house  from  December  1  to 
January  12,  at  $50  a  month.  Ans.  $68.33. 

4.  What  will  be  the  rent  of  a  house  from  January  20 
to  August  9,  at  $750  a  year,  payable  quarterly? 

Ans.  $416.67. 

The  Teacher  can  give  more  of  such  questions  as  he  finds  it  nec- 
essary. 

SERVANTS'  WAGES. 

Servants  are  hired  by  the  week  or  month  of  four  weeks 
or  calendar  month,  and  are  entitled  to  wages  every  day, 
Sunday  included. 


30       = 


276          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

* 

5.  A  girl  hires  on  September  3  and  leaves  on  October 

9;  what  will  be  her  wages  at  $3  a  week? 

From  September  3  to  October  9  is  36  days,  or  5  weeks 
1  day. 

Wages  for  5  weeks  at  $3=  15.00 
Wages  for  I  day=\  of  $3=       .428 

15.428 
or  $15.43 

6.  A  man  is  hired  on  June  9  at  $40  a  calendar  month, 
and  is  discharged  on  September  3;  what  is  the  amount  of 
his  wages?  Ans.  $113.33. 

This  is  computed  in  the  same  way  as  house  rent. 

7.  What  will  be  the  wages  of  a  man  for  7  months  and 
7  days  at  $33  a  month?  Ans.  $238.70. 

To  find  the  quantity  of  carpet  for  a  floor. 

Most  carpeting  is  made  one  yard  in  breadth.  Brussels 
and  velvet  carpeting  is  usually  made  only  f  of  a  yard  or 
27  inches,  though  sometimes  it  is  made  |-  and  even  f,  or 
double  the  usual  breadth. 

Oil  cloths  vary  in  breadth  from  3  to  24  feet,  as  follows : 
3  ft.  9  in.,  4  ft.  6  in.,  7  ft.  6  in.,  12  ft.,  18  ft.  and  24  ft. 

Matting  is  of  three  kinds:  China  Cocoa,  Manilla  and 
Cane.  China  matting  is  made  of  a  kind  of  rushes.  It- 
looks  neat  but  does  not  wear  long.  The  best  kinds  are 
Gowqua  and  Manning. 

Cocoa  matting  is  made  of  a  kind  of  grass.  The  best 
quality  is  called  "diamond  A,"  from  the  brand  found 
upon  it. 

Common  ingrain  carpeting  may  be  matched  by  cutting 
through  the  center  of  the  pattern  ;  but  expensive  carpets 
can  be  matched  only  by  persons  experienced  in  the  busi 
ness.  Some  of  them  require  two  webs,  others  more,  to 


HOUSEKEEPING.  277 

make  a  pattern.  Carpet  dealers  usually  furnish  their  car- 
pets made  to  any  dimensions,  and  even  lay  them  when  re- 
quired. 

The  quantity  of  carpet  required  for  a  room  is  found  by 
multiplying  the  length  by  the  breadth,  in  feet  or  inches, 
and  dividing  by  the  number  of  square  feet  or  inches  in  a 
yard.  For  f  carpet,  divide  the  square  feet  by  6-J;  for 
yard,  divide  by  9.* 

ANOTHER  WAY 

Is  to  find  the  number  of  Ireadths  required,  and  multiply 
it  by  the  length  of  the  room. 

8.  How  much  ingrain  carpet  will  be  required  to  cover 
a  room  15  by  20  feet? 

Cutting  the  carpet  in  its  greatest  length,  there  would 
be  5  breadths,  which,  multiplied  by  the  length,  gives  100 
feet,  or  33J  yards. 

Required  the  quantity  of  ingrain  carpet  to  cover  three 
rooms,  measuring  as  follows:  one  room  12  by  16  feet; 
one,  16  by  21;  one,  15  by  19;  and  one  room,  20  by  25 
with  velvet  carpet. 

9.  What  quantity  of  velvet  carpet  will   cover  a  saloon 
20  by  40  feet? 

The  breadth,  20  feet,  reduced  to  inches— 240,  which, 
divided  by  27  inches=9  breadths,  nearly.  The  length, 
40  feet,  multiplied  by  9^360  feet,  or  120  yards. 

for  calculations  pertaining  to  wall  paper,  see  Paper 
Hanging,  page  267. 

For  calculations  pertaining  to  shopping,  see  page  111. 

For  weights  and  measures,  see  Tables. 

*  The  spaces  for  fire-places,  etc.,  will  allow  sufficient  for  waste. 
Carpets  should  be  cut  a  few  inches  short  to  allow  for  stretching. 


278          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

To  find  the  cost  of  articles  sold  by  the  dozen. 

1  article  will  cost  y1^-  of  the  cost  of  a  dozen. 

2  articles  "       "      £    "     "       "       "         " 

g         u  it         it         l      u       u         it         it  it 

4  u  it         a        ^_     it       it         it         it  it 

5  "         "       "       5  times  the  twelfth. 

6  u         "       "      |  of  the  cost  of  a  dozen. 

8  "         "       "       2  times  the  third. 

9  "        "       "      3      "       "     fourth. 

10  "        "       "      $  off. 

11  "        "       "     -J.  " 


XXXVII.  GEOMETRICAL  DEFINITIONS. 

An  Angle  is  the  opening  between  two  lines  that  meet 
in  a  point. 

A  Right  Angle  is  made  by  one  straight  line  standing 
perpendicular  to  another. 

An  Obtuse  Angle  is  wider  than  a  right  angle. 

An  Acute  Angle  is  less  than  a  right  angle. 

A  Triangle  is  a  figure  having  three  sides  and  three 
angles. 

An  Equilateral  Triangle  has  all  its  sides  equal. 

An  Isoscetes  Triangle  has  two  of  its  sides  equal. 

A  Scalene  Triangle  has  all  its  sides  unequal. 

A  Right-Angled  Triangle  has  one  right  angle. 

An  Obtuse-Angled  Triangle  has  one  obtuse  angle. 

An  Acute-Angled  Triangle  has  all  its  angles  acute. 

A  Quadrangle  or  Quadrilateral  is  a  four-sided  figure, 
and  may  be 

A  Parallellogram,  having  its  opposite  sides  parallel ; 

A  Rectangle,  having  four  right  angles,  sides  unequal; 


GEOMETRICAL  DEFINITIONS.  279 

A  Square,  having  all  its  sides*  equal,  and  its  angles 
right  angles; 

A  Rhombus  or  Lozenge,  having  its  sides  equal  and  no 
right  angle; 

A  Rhomboid,  a  parallelogram,  with  no  right  angles ; 

A  Trapezium,  having  unequal  sides; 

A  Trapezoid,  having  only  two  sides  parallel. 

Polygon,  a  plain  figure  having  more  than  four  sides. 

A  Pentagon  has  nre  sides,  a  hexagon  six,  a  heptagon 
seven,  an  octagon  eight,  a  nonegon  nine,  a  decagon  ten,  etc. 

A  Circle  is  a  plain  figure,  bounded  by  a  curved  line,  all 
points  of  which  are  equidistant  from  the  center. 

An  Arc  is  any  part  of  a  circumference. 

A  Chord  is  a  straight  line  joining  the  extremities  of  an 
arc. 

A  Segment  of  a  circle  is  a  part  of  a  circle  bounded  by 
an  arc  and  its  chord. 

The  Radius  of  a  circle  is  a  line  extending  from  the 
center  to  the  circumference. 

A  Quadrant  is  a  quarter,  a  sextant  a  sixth  of  a  circle. 

A  Zone,  a  part  of  a  circle  included  between  two  parallel 
chords. 

A  Prism  is  a  solid,  the  sides  of  which  are  parallelo- 
grams. It  may  have  three  or  more  sides. 

A  Pyramid  is  a  solid  with  regular  sides,  tapering  to  a 
point. 

A  Cylinder  is  a  solid  of  uniform  thickness,  having  its 
ends  circular. 

A  Cone  is  a  round  pyramid,  having  a  circle  for  its  base. 

The  Circumference  of  a  circle  is  the  line  by  which  it  is 
bounded.  The  Diameter  is  a  line  drawn  through  the  cen- 
ter and  terminating  at  the  circumference.  The  Radius  is 
a  line  drawn  from  the  center  to  any  point  in  the  circum- 
ference. 


280 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


j 

—  — 

— 

"1 

,  —  j.  — 

I 

j 

XXXVIII.  MENSURATION. 

315.  To  find  the  area  of  a  square,  multiply  the  length 
by  the  breadth. 

1.  Let  one  side  of  the  annexed  parallelo- 
gram be  3,  and  the  other  4,  then  the  area 
will  be  3X4=12. 

2.  What  is  the  area  of  a  square,  the  side 
of  which  is  3  feet  6  inches.  Ans.  12J. 

316.  To  find  the  area  of  a  Rhombus  or  Rhomboid,  we 
multiply  the  length  by  the  perpendicu- 
lar height.     The  reason  for  this  will  be 

evident  from  an  inspection  of  the  fig- 
ures. The  triangle  A  C  m  of  the  rhom- 
bus applied  to  the  side  B  D,  will  make 
a  square ;  and  the  triangle  of  the  rhom- 
boid applied  to  the  other  side  will  make 
a  parallelogram. 

3.  Let  the   perpendicular   height   (A 
m)  be  20,  and  the  length  (A  B  or  C  D) 
be  30,  then  the  area  will  be  20x30—600. 

4.  What  will  be  the  area  of  a  rhomboid,  the   perpen- 
dicular and  length  being  15  and  25?  Ans.  375. 

317.  To  find   the   area   of  a  right-angled   triangle,  we 
multiply    the    perpendicular    by   half  B 
the  base,  or  the  base  by  half  the  per- 
pendicular. 

REASON. — The  part  A  m  n,  applied  to  the      j 
part  B  C  m  n,  with  the  line  A  m  applied  to  m 

B,  and  the  point  A  on  the  point  B,  will  make  a  parallelogram,  with 
a  base  half  of  that  of  the  triangle. 


MENSURATION.  281 

5.  Let  the  base  be  10  and  the  perpendicular  8;  then 
1£X  8=40,  the  area. 

6.  The  perpendicular  is  16  and  the  base  120;  what  is 
the  area?  Ans.  960. 

318.  When  the   triangle  is  not  right-angular,  half  the 
base   multiplied    on   the    height  will 

give  the  area. 

REASON.—  The  triangle  C  A  D  is  half  the 
rhomboid. 

7.  Let  the  base  be  30  and  the  height  20,  then  20  X-3/— 
300,  the  area. 

319.  When  the  perpendicular  is  not  given,  the  area  can 
be  found  by  subtracting  each  side  from  half  of  the  sum  of 
the  sides;  then  by  multiplying  these  three  remainders  and 
half  the  sum   of  the   sides  together,  and   extracting  the 
square  root  of  the  product 


8.  Let  the  sides  be  5,  7  and  10  ;  then  -  -  ---  —^— 


11—  7=4 

11—10=1  6X4X11=264,  the  sq.  root  of  which  is    16.3. 

9.  What  is  the  area  of  a  triangle,  the  sides  of  which  are 
50,  30  and  40?  Ans.  600. 

320.  To  find  the  area  of  a  circle,  multiply  the  square 
of  the  diameter  by  .7854. 

Multiply  half  the  circumference  by  half  the  diameter. 

321.  To  find  the  side  of  a  square  equal  in  area  to  a  given 
circle,  multiply  the  circumference  by  .2820948,  or  the  di- 
ameter by  .8862269. 

10.  The  diameter  of  a  circle  is  100;  required  the  side 
of  a  square  having  the  same  area. 

.8862269X100=88.62269,  side  of  a  square, 
24 


282 


NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


322.  To  find  the  side  of  an  inscribed  square,  multiply 
the  diameter  by  .7071068. 

11.  The  diameter  of  a  tree  is  2  feet;  re- 
quired the  side  of  a  square  log  that  may 
be  cut  from  it? 

.7071068X2=1.4142136,    or    1    foot    5 
inches  nearly. 

323.  Having  the  side  of  a  square,  to  find  the  diameter  of 
a  circumscribed  circle,  multiply  the  side  by  1.4142136. 

324.  From  the  side  of  a  square  to  find  the  circumference 
of  a  circumscribed  circle,  multiply  the  side  by  4.4428934. 

325.  To    find    the    diameter,    multiply    the    side     by 
1.1283791. 

326.  From  the  side  of  a  square,  to  find  the  circumference 
of  a  circle  of  equal  area,  multiply  the  side  by  3.5449076. 

327.  To  find  the  area  of  a  trapezium,  we  divide  it  into 
two  triangles,  and  the  sum  of  the  areas 

will  be  the  area  required. 

12.  Let   the   diagonal   A   C   be   100, 
and  the  perpendiculars  B  m  and  D  n: 
30  and   35  ;   then 

(35+30)  X 100 

V     ^     ' =3250,  the  area. 

REMARK.— The  areas  of  irregular  polygons  are  found  by  dividing 
the  figures  into  triangles,  and  taking  the  sum  of  their  areas. 

328.  To  find  the  circumference  of  a  circle,  we  multiply 
the  diameter  by  3.1416,  or  3£,  because 

the  circumference  of  a  circle  is  3^  times 
greater  than  the  diameter. 

13.  Let  the  diameter  be  5;  then  3.1416 
X 5=15. 708,  the  circumference. 

The  diameter  is  found  by  dividing   the  circumference 
by  3.1416. 


MENSURATION. 


283 


329.  To  find  the  area  of  a  regular  polygon,  add  all  the 
sides  together,  and  multiply  by  the  perpendicular  drawn 
from  the  center  of  the  polygon  to  the  middle  of  one  of  its 
sides;  or, 

Multiply  the  square  of  the  side  of  the  polygon  by  the 
number  standing  opposite  to  the  number  of  its  sides  in 
the  following  table: 


3 

0.4333012 

8 

4.8284271 

4 

1. 

9 

6.1818242 

5 

1.7204774 

10 

7.6942088 

6 

2.5980762 

11 

9.3656404 

7 

3.6339124 

12 

11.1961524 

14.  The  side  of  a  pentagon   (a  five-sided  figure)  is  20 
yds.  and  its  perpendicular  13.76382;  required  the  area. 

20X5X13.76382 
First  method, =688.191,  Ans. 

Second  method,  202X  1.720477=688.19,  Ans. 

15.  The  side  of  a  nonegon  is  50  inches;   required  its 
area.  Ans.  15454.5605. 

330.  To  measure  the  heights  of  objects,  the  tops  of  which 
can  not  be  reached,  the  shadow  cast  by  the  tree  may  be 
used.  Measure  the  length  of  the  shadow  cast  by  the  ob- 
ject, and  that  of  some  object  the  length  of  which  is 
known;  then  the  shadow  of  the  known  object  will  be  to 
that  of  the  first  as  the  length  of  the  known  object  to 
the  length  of  the  first 

16.  Let  the  known  object  be  a  man,  who,  without  his 
hat,  measures  (in  his  shoes)  5  feet  8  inches,  and  whose 
shadow  measures  15  feet,  while  the  shadow  of  a  tree  meas- 
ures 120  feet. 

15  ;  120  :  :  68  inches  :  the  height  of  the  tree  in  inches. 

120X68 

=544  inches,  or  45J  feet. 


284          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

MEASUEEMENT  OF  SOLIDS. 

331. .  To  find   the  solidity  of  a  cube,  multiply  the  side 
by  itself,  and  the  product  again  by  the  side. 

17.  The  side  of  a  cubical  block  of  marble  is  5  feet  7 
inches;  what  is  the  solid  contents? 

5_7^x5T7.TX5T72=174  feet,  nearly. 

332.  To  find  the  solidity  of  a  parallelopipedon  (a  solid 
figure  with  square  corners),  multiply  the  length,  breadth 
and  thickness  together. 

18.  A    log    measures   7  feet  in   length   and  15   by    20 
inches  in  thickness;  required  the  solid  contents. 

7X15X20=2100  square  inches,  or  14T7^  feet. 

333.  To  find  the  solidity  of  a  prism,  multiply  the  area 
of  the  end  by  the  length. 

19.  What  is  the    solidity  of  a  prism   whose   ends   are 
equilateral   triangles,   each   side   of  which   is   4  feet  and 
height  8  feet? 

An  equilateral  triangle  is  made  up  of  two  right-angled 
triangles,  the  perpendicular  of  which  is 
found  by  taking  the  square  root  of  the 
difference  of  the  squares  of  half  the  base 
and  the  other  side. 


y/42— 22=3.464,     the    perpendicular, 
m  n. 

3.464x2=6.928=area  of  the  end. 

6.928X8=55.424  feet,  the  solidity. 

334.   To  find  the  solidity  of  a  cone,  multiply  the  area 
of  the  base  by  one-third  the  height. 

20.  A  cone  is  10  feet  in  diameter  and  10  feet  high;  re- 
quired the  solidity. 

102X -7854=78.54,  area  of  the  base,  which,  multiplied 
by  J  the  height,  3J=261.8,  the  solidity. 


MENSURATION.  285 

21.  How  many  cubic  feet  in  a  cone  whose  diameter  is  12 
feet,  and  its  perpendicular  height  100?        Ans.  3769.92  ft. 

335.  To  find  the  solidity  of  a  pyramid,   multiply   the 
area  of  the  base  by  one-third  the  height. 

22.  A  square  pyramid  has  a  base  of  4  feet  and  height 
of  12  feet;  required  the  solidity. 

4x4  —area  of  base. 
16X^=64  for  the  solidity. 

23.  The  spire  of  a  church  is  an  octagonal  pyramid,  each 
side  at  the  base  being  5  feet  10  inches,  and   its  perpen- 
dicular  height  45   feet;  also   each   side   of  the  cavity  or 
hollow  part  at  the  base  is  4  feet  11  inches,  and  its  per- 
pendicular height  41  feet ;  how  many  solid  yards  of  stone 
does  the  spire  contain?  Ans.  32|-,  nearly. 

336.  To  find  the  solidity  of  the  frustrum  of  a  cone  or 
pyramid* 

Find  the  sum  of  the  areas  of  the  two  ends,  and  of  a 
geometrical  mean  between  them,  and  multiply  by  one- 
third  the  perpendicular  height. 

24.  What   is    the    solid    contents    of    a   frustrum    of   a 
square   pyramid,  whose   sides   are   5   and  3,  and  perpen- 
dicular height  12? 

52       X   .7854=19.6350  area  base. 
32       X   .7854=  7.0686  area  upper  end.   . 
1/r9.635x7.0686=11.7810  geometrical  mean. 
38.4846 

4  one- third  the  height. 
153.9384 

25.  What  is  the  solidity  of  a  squared  piece  of  timber, 
its  length  being  18  feet,  and  sides  of  the  bases  18  and  12 
inches?  Ans.  28.5  ft. 

*A  segment  is  a  piece  cut  off  by  a  plane,  parallel  to  the  base;  a 
fruslrwn  is  what  remains  at  the  base. 


286          NELSON'S  COMMON-SCHOOL  ARITHMETIC. 

26.  How  many  cubic  feet  of  timber  in  a  tapering  log 
14.25  ft.  long,  diameters  9  and  18  in.?         Am.  14.689  ft. 

Comparison  between  the  globe,  cylinder  and  cone,  the  di- 
ameter and  heights  being  100: 

Solid  contents  of  the  cylinder,  785.4 
"  "  "        sphere,     523.6=f  of  the  cylinder. 

"  "  "        cone,        261.8=4      u  " 

REMARKS. — 1.  The  cone  cut  out  of  a  solid  cylinder,  whose  diame- 
ter and  height  are  equal,  will  leave  a  part  equal  to  the  solidity  of 
a  sphere  of  the  same  diameter. 

2.  A  square  pyramid,  whose  height  and  side  are  equal  to  the  side 
of  a  cube,  if  cut  out  of  the  latter,  will  leave  f  of  the  cube. 

GAUGING. 

The  process  of  finding  the  capacity  of  barrels,  etc.,  is 
called  gauging. 

337.  llfiving  the  head  and  Lung  diameter  and  the  di- 
ameter between  them,  to  find  the  capacity  of  a  barrel  or 
cask  in  gallons,  we  add  together  the  square  of  the  head 
and  bung  diameters  and  twice  the  middle  diameter,  and 
multiply  the  sun;  by  the  length,  and  that  by  .0004721  for 
imperial  gallon. 

27.  A  cask,  having  for  head,  bung  and  middle  diameter 
30,  36  and  33,  and  length  40  inches,  holds  how  many  im- 
perial gallons? 

302-f362-f(33x2)X40x. 0004721=42.72  galls. 

Practical  method  for  measuring  small  cylindrical  vessels, 
is  to  multiply  the  square  of  the  diameter  by  34,  and  that 
by  the  height  in  inches,  and  point  off  four  figures.  The 
result  will  be  the  capacity  in  gallons. 

28.  An  oil-can  measures  12  inches  in   diameter  and  2 
feet  in  height;  required  the  contents  in  gallons. 

122X34X 24=117504=11.75  or  11$  galls. 


THE   METRIC   SYSTEM.  287 


XXXIX.   THE   METRIC   SYSTEM. 

338.  The  Metric  System  is  a  decimal  system  of  weights 
and  measures,  of  French  origin,  deriving  its  name  from 
Meter,  the  unit  of  measure  upon  which  the  system  is  based. 

Since  1840  it  has  been  adopted  by  most  European  gov- 
ernments, including  that  of  Great  Britain  in  1864,  and 
has  been  in  use  by  men  of  science  every-where.  During 
its  last  session  (39th),  Congress  authorized  its  use  in  this 
country,  and  made  provision  for  its  immediate  introduc- 
tion into  post-offices.* 

339.  The  units  of  measure  are  the  meter,  are,  liter,  and 
stere;  and  the  unit  of  weight  the  gram.     Other  denomina- 
tions are  formed  from  these  by  prefixing  Greek  or  Latin 
numerals;  the  former  for  denominations  above  the  unit, 
and  the   latter  for  denominations   below  the  unit.     The 
Greek  prefixes  are  deka,  10;  hecto,  100;   kilo,  1000;  and 
myria,  10000.     A  dekameter  is  ten  times  and  a  hectome- 
ter one  hundred  times  the  length  of  a  meter.     The  Latin 

*A  Bill  to  authorize  the  use  of  the  Metric  System  of  Weights  and 

Measures. 

J3e  it  enacted  by  the  Senate  and  House  of  Representatives  of  the 
United  States  of  America,  in  Congress  assembled,  That  from  and  after 
the  passage  of  this  act,  it  shall  be  lawful  throughout  the  United 
States  of  America  to  employ  the  weights  and  measures  of  the 
Metric  System;  and  no  contract,  or  dealing,  or  pleading  in  any 
court,  shall  be  deemed  invalid  or  liable  to  objection  because  the 
weights  or  measures  expressed  or  referred  to  therein  are  weights 
or  measures  of  the  Metric  System. 


288        NELSON'S   COMMON-SCHOOL   ARITHMETIC. 

prefixes  are  deci,  TVth;  CentL  TJ-g-th;  milli,  y^Vu*^-  A 
decimeter  is  one-tenth  and  a  centimeter  one-hundredth  of 
the  length  of  a  meter. 


LIST   OF    NAMES   AND    THEIR   PRONUNCIATION. 


Name. 

Pi'onunciation. 

Abbreviation. 

Meter, 

Me'ter, 

M. 

Kilometer, 

Kil'o-meter, 

K.  M, 

Hectometer, 

Hec'to-meter, 

H.  M, 

Dekameter, 

Dek'a-meter, 

D.  M, 

Decimeter, 

Des'i-meter, 

d.  m. 

Centimeter, 

Cent'i-ineter, 

c.  m. 

Are, 

Aer, 

A. 

•  Hectare, 

Hector, 

H,  A. 

Centare. 

Seut'iir, 

c.  a. 

Stere, 

Stere, 

S. 

Dekastere, 

Deckl-stere, 

D.  S. 

Decistere, 

Des'i-stere, 

d,  s. 

Gram, 

Gram, 

G. 

Dekagram, 

Dekxa-gram, 

D.  G. 

Kilogram, 

KiFo-gram, 

K.  G. 

Hectogram, 

Hec^o-gram, 

H.  G. 

Decigram, 

Des^-gram, 

d.g. 

Centigram, 

Sent^-gram, 

e.g. 

Milligram, 

Mil'li-gram, 

m.  g. 

Tonneau, 

Tonxno, 

Ton. 

Millier, 

MiKli-er, 

Mil. 

Liter, 

L^ter, 

L. 

Kiloliter, 

KiKo-k-ter, 

K.  L. 

Hectoliter, 

Ilect/o-le-ter, 

H.  L. 

Dekaliter, 

Dekxa-te-ter, 

D.  L. 

Deciliter, 

Des/i-le-ter, 

d.  1. 

Centiliter, 

SenKi-le-ter, 

c.  1. 

Milliliter, 

MiKli-lg-ter, 

m.  1. 

NOTE.  —  Denominations 

below  units  arc 

abbreviated  with  small 

letters. 

THE  METRIC  SYSTEM.  289 


MEASURES   OF   LENGTH. 

840.  Besides  its  being  the  base  of  the  new  system,  the 
Meter  is  the  unit  of  measure  for  lengths,  and  is  one  ten- 
millionth  part  of  the  distance  from  the  Equator  to  the  poles, 
and  is  equivalent  to  39.37  inches  ordinary  measure. 


Myriameter  —  10,000  Meters  or  393685  inches. 

Kilometer  —  1,000  "  or  39368.5  " 

Hectometer  =  100  "  or  3936.85  " 

Dekameter  —  10  <•  or  393.685  " 

Meter  =  1  «  or  39.3685  " 

Decimeter  =  T^th  "  or  3.9368  " 

Centimeter  ==  ^th  "  or  .39368  " 

Millimeter  —  T?rWth  "  or  .03936  « 


REMARK. — Meters,  when  combined  with  lower  denominations^ 
are  treated  as  whole  numbers,  and  the  latter  as  decimals. 

The  denominations  most  used  are  the  Kilometer,  Centimeter^ 
and  Millimeter.  7  Myriameters,  8  Kilometers,  and  6  Hectome* 
ters  would  be  written  78.5  K.  M. 

SQUARE   OR   SURFACE   MEASURE. 

341.  The  unit  of  measure  for  large  surfaces  is  the  Are, 
from  which  are  derived  the  Hectare  and  Centare.  For 
smaller  surfaces  the  denominations  are  the  same  as  for 
measures  of  length,  with  the  addition  of  the  word  square. 

Centare  =          1  sq.  meter,  or  1560  sq.  inches. 

Are         =       100  sq.  meters,  or  1  sq.  dekameter,  or  119.6  sq.  yds. 

Hectare  =  10,000  sq.  meters,  or  1  sq.  hectometer,  or  2.471  acres. 

1  sq.  decimeter,  d.  w.,2    =  TJ^  sq.  meter 

1  sq.  centimeter,  c.  m.,2  =-.  TJ^  sq.  decimeter,  or  TQ JQQ  sq.  meter. 

1  sq.  millimeter,  m.  m.,2  =  TJ^  sq.  centimeter,  or  T^^J^^^  sq.  c.  m. 

27.354  M.2  =  27  sq.  meters,  35  sq.  decimeters,  40  sq.  centimeters. 


290        NELSON'S    COMMON-SCHOOL  ARITHMETIC. 


CUBIC   OR   SOLID   MEASURE. 

342.  The  Stere  may  be  called  the  unit  for  cubic  meas- 
ure.    It  is  equal  to  a  cubic  meter  or  1.308  yards. 

1  dekastere,  D.  S.,=W  steres,  or  13.08  yards. 

1  decistere,  d.  s.,—  TV  stere,  or  0.1308  yards. 

1  cubic  decimeter,  d.  m.f  =T^Vtf  cubic  meter. 

1  cubic  centimeter,  c.  m.^=T^inf  cubic  decimeter,  or  -j-^J^^  M. 

1  cubic  millimeter,  m.wi.,=T^  cubic  centimeter,  or 


The  Stere,  etc.,  is  used  for  measuring  fire-wood  and 
lumber,  and,  for  computing  large  numbers,  is  preferred 
to  the  other  denominations. 

MEASURES   OF   CAPACITY. 

343.  The  Liter  is  the  unit  of  measure  for  capacity, 
and  is  equal   to  a  cubic  decimeter  or  1.0567  quarts  of 
United  States  liquid  measure. 

TABLE. 

Kiloliter     =  1,000  liters,  or  1  cubic  meter  of  water. 
Hectoliter  =     100      "      or  ^  cubic  meter. 
Dekaliter  =       10      "       or  10  cubic  decimeters. 
Liter  of  water  weighs  1  kilogram. 
Deciliter    =   j1^   liter,  or  ^  cubic  decimeter. 
Centiliter  =  T^  liter,  or  10  cubic  centimeters. 

A  milliliter  of  water  weighs  a  gram.  The  liter  and 
hectoliter  are  most  in  use. 

WEIGHTS.     , 

344.  The  Gram  is  the  unit  of  weight,  and  is  equal  to 
15.432  grains  Troy,  which  is  the  weight  of  a  cubic  centi- 
meter of  pure  water  at  its  greatest  density. 


THE   METRIC   SYSTEM. 


291 


Kilogram     =  1,000  grams. 
Hectogram  =     100       " 
Dekagram  =        10       " 
Gram  —          1    gram. 

Decigram    =       •£§       " 


Centigram  =  ^  gram. 
Quintal       =    100  kilograms. 
Tonneau      —1,000  kilograms, 
or  2,204  Ibs. 


REMARK.  —  The  gram  and  its  subdivisions  are  used  in  com- 
pounding medicines,  and  wherever  great  accuracy  is  required. 

The  kilogram  is  the  denomination  most  used,  and  weighs  a 
little  over  2£  pounds.  The  quintal  and  tonneau  are  used  for 
heavy  weights,  but  may  be  expressed  in  kilograms. 


COMPARISON  OF  METRIC   DENOMINATIONS  WITH  THOSE  IN 
PRESENT    USE. 

MEASURES  OF  LENGTH. 


NAMES. 

VALUES. 

EQUIVALENTS 

IN    USE. 

Mvriameter                            .. 

10  000  meters  

6.2137    miles. 

Kilometer  

1,0(10       ik      

0.62137      " 

100        "       

328  l-l2th  feet. 

Deka  meter  

10        "      

393.7     inches. 

39  37          " 

Decimeter  
Centimeter  
Millimeter  

l-10th  of  a  meter  
1  -100th  of  a  meter  
1-loooth  of  a  meter  

3.937        " 
0.3937      " 
0.0394       ** 

MEASURES  OF  SURFACE. 


NAMKS. 

VALUES. 

EQUIVALENTS  IN  USE. 

Hectare  10,000 

square  meter.*  

2.471  acres. 

A  re                                           100 

Con  tare  1 

ki       meter  

l">f>0         square  indies. 

MEASURES  OF  CAPACITY. 


NAMES. 

No.  of 
Liters. 

CUBIC  MEASURE. 

DRY  MEASURE. 

LIQUID  OR 
WINK  ME  AS. 

Kiloliter  or  Stere. 
Hectoliter  

1,000 
100 

cubic  met«-r  
-10th  cubic  meter 

1.3()<S  cubic  yards... 
2  bus   &  3  3r>  pecks 

264.  17  grails. 
26.417  £alls. 

Dekaliter 

10 

9  og  quart* 

2  6417  galls 

Liter 

1 

cubic  decimeter 

0  'MIX  quart 

I  ur>67  quarts. 

Deciliter  
Centiliter  
Milliliter  

1-10 
1-100- 
1-1000 

-1<>  cubic  decimeter. 
0  cubic  centimeters 
cubic  centimeter... 

6.  1022  cubic  inches. 
0.6102  cubic  inch  ... 
0.06102  cubic  inch.. 

O.S45  trill. 
0.:W8  fluid  oz. 
0.27  fluid  drin. 

292         NELSON'S  COMMON-SCHOOL  ARITHMETIC. 


WEIGHTS. 


NAMES. 

No.  OF 
GRAMS. 

WEIGHT  IN  QUANTITY  OF 
WATER   AT   MAXIMUM 
DENSITY. 

EQUIVALENTS 

IN   USE. 

Millinr  or  Toiine.au... 

i  000  <H)0 

1  cubic  meter                 

2204  6  pounds  Av. 

Quintal  

1(1(1  Off!  I 

220.46        " 

10  (too 

22.046        " 

1,(MXJ 

1  liter  

2.2046        *' 

Hectogram  

100 

1  deciliter   .                     

3.5274  ounces 

Dekagram  

10 

10  cubic  centimeters  

0  3527  ounce 

1 

1  cubic  centimeter  

15.'432  grains  T  oy. 

1-10 

1.5432       " 

Centigram  
Milligram  

1-100 
1-llXJO 

10  cubic  millimeters  
1  cubic  millimeter  

0.1543  grain 
0.0154       " 

CALCULATIONS   PERTAINING   TO   THE    METRIC   SYSTEM. 

The  principal  recommendation  of  the  new  system  is  the 
simplicity  of  its  arithmetic.  Under  it  the  long  and  in- 
tricate calculations  of  the  present  system  are  unknown. 
Denominations  are  changed  from  one  to  another  by  the 
use  of  the  decimal  point,  with  sometimes  one  or  more 
ciphers,  while  addition  and  other  operations  are  per- 
formed precisely  as  in  whole  numbers  and  decimals. 

1.  In  35.2  D.  M.  how  many  M?     Ans.  352  M. 
Solution.— In  1  D.  M.  there  are  10  M.,  in  35.2  D.  M. 

there  are  35.2  times  10  or  352  M. 

2.  In  35142  millimeters  how  many  meters? 

Ans.  35.142  M. 

Solution. — 1000  m.  m.  make  1  M.,  hence  we  divide  by 
1000,  which  gives  35.142  M. 

3.  In  31475  M.  how  many  kilometers? 

Ans.  31.475  K.  M! 

4.  Reduce  371.2  D.  M.  to  meters.  Ans.  3712  M. 

5.  lleduce  45.67  M.  to  decimeters.        Ans.  456.7  d.  c. 

6.  lleduce  13456  decimeters  to  meters.    Ans.  1345.6  M. 

7.  In  213.21  M.  M.  how  many  dekameters? 

Ans.  213210  D.  M. 

8.  lleduce  3157.2  c.  m.  to  meters.          Ans.  31.572  M. 


THE   METRIC   SYSTEM.  293 

9.  A  man  traveled  80.5  K.  M.  in  a  day,  how  far  would 
he  travel  in  5  days  at  the  same  rate  ?     Ans.  152.5  K.  M. 

10.  A  ship  steams  15  K.  M.  in  an  hour,  how  long  would 
it  take  her  to  go  a  distance  of  50  myriameters  ? 

Ans.  33^  hours. 

11.  At  2  cents  per  kilometer,  what  would  be  the  cost 
of  traveling  351.27  M.  M.  ?  Ans.  $70.25. 

12.  A  man  walked  a  distance  of  351.27  K.  M.  in  9  days, 
how  many  was  that  per  day?  Ans.  39.03  K.  M. 

13.  In  253  d.  m.,  7  c.  m.,  5  m.  m.,  how  many  meters  ? 

Ans.  25.375  M. 

14.  In  157  d.  m2.,  35  c.  m2.,  how  many  square  meters  ? 

Ans.  1.5735  M2. 

15.  At  15  cents  per  M2.,  what  would  25  H.  M2.  of  lum- 
ber cost?  Ans.  $375. 

16.  How  many  square  meters  in  a  floor  which  measures 
6  M.  long  and  5.25  M.  broad?  Ans.  31.5  M2. 

17.  A  pavement  measures  39.2  M.  long  and  4.15  M. 
broad,  how  many  square  M.  does  it  contain? 

Ans.  162.68  M.a 

18.  A  strip  of  land  120  M.  long  contains  342  ares, 
what  is  its  breadth?  Ans.  285  M. 

19.  What  will  545.37  centares  cost  at  $4.15  per  are  ? 

Ans.  $22.63. 

20.  From  1374.27  M2.  take  7.14  ares. 

Ans.  6.6027  M2. 

21.  At  $50.50  per  H.  A.,  what  will  1371.154  ares  cost? 

Ans.  $692.43. 

22.  In  37.27  M3.  how  many  decisteres?     How  many 
dekasteres?  Ans.  372.7  d.  s.,  3.727  D.  S. 

23.  At  $1.50  per  M3.,  what  will  157.28  dekasteres  cost? 

Ans.  $2359.2. 


294        NELSON'S   COMMON-SCHOOL   ARITHMETIC. 

24.  What  will  153  S.  of  wood  cost  at  $1.75  per  S.? 

Ans.  $267.75. 

25.  $875.55  was  paid  for  413.5  S.  of  wood,  what  was 
the  cost  per  stere?  Ans.  $2.17. 

26.  A  cistern  is  3  M.  long,  2  M.  wide,  and  1  M.  deep, 
what  weight  of  water  will  it  contain? 

Ans.  13227.6  Ibs.,  or  6  tonneau. 

27.  What  will  a  decistere  of  wood  cost,  at  $3.57  per 
stere?  Ans.  36  cents. 

28.  Find  the  price  of  53  liters  of  wine,  at  $3.50  per 
dekaliter.  Ans.  $18.55. 

29.  A  vessel  measures  53.5  liters,  what  weight  of  water 
does  it  contain?  Ans.  53.5  kilograms. 

30.  In  .3514  H.  L.  how  many  liters?     Ans.  35.14  L. 

31.  In  145.37  M3.  how  many  liters?     Ans.  145370  L. 

32.  Find  the  price  of  5.3  liters  of  wine,  at  $1.75. 

Ans.  $92.75: 

33.  At  57  cents  a  meter,  what  will  317.5  meters  cost? 

Ans.  $180.975. 

34.  An  are  of  land  cost  $53,  what  was  the  price  per 
meter?  Ans.  53  cents. 

35.  A  kilogram   costs   $37.50,  what  will  a  dekagram 
cost?  Ans.  37-J-  cents. 

36.  How  many  ares  in  a  square  piece  of  land  which 
measures  150  meters  in  length  and  500  in  breadth? 

Ans.  750. 

37.  At  $25  a  kilogram,  what  will  a  dekagram  cost? 

Ans.  25  cents. 

38.  At  $7.25  a  stere,  what  will  320  steres  of  wood  cost? 

Ans.  $2320. 


THE   METRIC   SYSTEM.  295 

343.  To  reduce  the  denominations  of  the  ordinary  to 
those  of  the  Metric  System. 

1.  In  3  miles  35  rods,  how  many  meters? 

3  miles  35  rods=197010  inches,  which,  divided  by  39.37, 
(the  number  of  inches  in  a  meter,)  gives  5004.06  meters. 

2.  In  5^  yards  how  many  meters?  Ans.  4.877. 

3.  Reduce  37 |~  feet  to  meters.  Ans.  11.43. 

4.  In  29  inches  how  many  centimeters?    Ans.  73.66. 

5.  In  6  inches  how  many  millimeters?      Aus.  152.4. 
61  How  many  square  meters  in  57  rods? 

Ans.  1441.68. 

7.  In  5  acres  how  many  ares?  Ans.  202.34. 

8.  How  many  liters  of  wine  in  37  gallons? 

Ans.  140.06. 

9.  In  57  yards  of  carpet  how  many  meters? 

Ans.  52.12. 

10.  How  many  liters  in  5^-  bushels?          Ans.  193.82. 

11.  How  many  cubic  meters  in  3759  cubic  feet? 

Ans.  106.44 

12.  Reduce  3  tons  to  tonneaus.  Ans.  2.72. 

13.  At  $2.50  a  liter,  what  will  20  gallons  cost? 

Ans.  $189.27 

14.  At  75  cents  a  meter,  what  will  135  yards  cost? 

Ans.  $92.58. 

344.  To  reduce  denominations  of  the  Metric  to  those 
of  the  ordinary  system. 

1.  In  175  meters  how  many  yards? 

Ans.  191  yds.,  1  ft.,  If  in. 

In  1  meter  there  are  39.37  inches,  in  175  meters  there 
are  175  times  as  many.  39.37X175=6889.75  inches, 
which,  reduced  to  yards,  rr=191  yards,  1  foot,  ly^  inches. 

2.  In  379.53  meters  how  many  yards?     Ans.  415.058. 


296        NELSON'S   COMMON-SCHOOL   ARITHMETIC. 

3.  Reduce  743.5  K.  M.  to  miles.  Ans.  461.988 

4.  In  2435  millimeters  how  many  inches? 

Ans.  95.866.. 

5.  Reduce  50  pounds  to  kilograms.  Ans.  22.68. 

6.  At  $2.75  per  kilogram,  what  will  375  Ibs.  cost? 

Ans.  $467.77, 

7.  At  35  cents  per  lb.,  what  is  it  per  kilogram? 

Ans.  77  cents. 

8.  At  $10.50  per  cord,  what  is  it  per  stere? 

Ans.  $2.90. 

9.  At  $1.25  a  bushel,  what  should  wheat  be  sold  at 
per  liter?  Ans.  3^  cents. 

10.  A  bushel  of  oats  weighs  33  pounds,  what  should 
it  weigh  per  liter?  Ans.  1.14  Ibs. 

11.  How  long  will  it  take  a  man  to  travel  a  distance  of 
57  K.  M.  who  walks  at  the  rate  of  3^  miles  per  hour? 

Ans.  10.119  hours. 

The  following  multipliers,  taken  from  a  pamphlet  writ- 
ten by  Prof.  H.  A.  Newton,  of  Yale  College,  will  facilitate 
the  labor  of  converting  the  denominations  of  one  system 
into  those  of  the  other.  They  will  be  found  sufficiently 
correct  for  ordinary  purposes : 

MetersX39.3685=inches.     InchesX.0254=nneters. 
MetersX3.2807=feet.     FeetX-30481—  meters. 
MetersXl-09357=yards.     YardsX.91444=meters.  • 
MetersX-19883r=:rods.    RodsX5.0294=meters. 
KilometersX-62135=miles.     MilesXl-6094=kilometers. 
Sq.  metersX!550— sq.  inches.     Sq.  inchesX.0006452=sq.  M. 
Sq.  metersX10.763=rrsq.  feet.     Sq.  feetX-09291— sq.  meters. 
Sq.  metersXl-196— sq.  yards.     Sq.  3'ardsX.8362=sq.  meters. 
AresX3.953=sq.  rods.     Sq.  rodsX.2529^ares. 
HectaresX2.4709=racres.    AcresX.4047=hectares. 


THE   METRIC   SYSTEM.  297 

Hect.aresX-003861=sq.  miles.     Sq.  milesX259.=rhectares. 
LitersX33.81— fluid  ounces.     Fluid  ouncesX-02958— liters. 
LitersXl-05656=quarts.     QuartsX-9465r=liters. 
LitersX.26414:=gallons.     GallonsX3.786=liters. 
HectolitersX2.837=bushels.     BushelsX.3524=hectoliters. 
LitersX61.012— cubic  inches.     Cubic  inchesX.01639=rliters. 
HectolitersX3.531=cubic  feet.     Cubic  feetX.2832=rhectoliters. 
SteresXl-3078— cubic  yards.     Cubic  yardsX-7646— steres. 
SteresX-2759=cords.     CordsX3.625z=Steres. 
GramsXl5-44— grains.     GrainsX-0648=grams. 
KilogramsX32.147--=troy  ounces.     Troy  oz.X-03108=rK.  G. 
KilogramsX35.30=avoirdupois  oz.     Av.  oz.X-02833^K.  G. 
KilogramsX2.681=Troy  pounds.     Troy  poundsX-373=K.  G. 
KilogramsX2.206=Av.  pounds.     Av.  poundsX4536=K.  G. 
TonneausX-985=long  tons.     Long  tonsXl-015=:tonneaus. 
TonneausXl-103=short  tons.     Short  tonsX-9066— tonneaus. 

1.  In  37  meters  how  many  yards? 

Ans.  37X1.09357=40.462  yds. 

2.  In  40.462  yards  how  many  meters? 

Ans.  40.462 X. 91444=37  meters. 

3.  In  120  kilograms  how  many  pounds  Avoirdupois? 

Ans.  120X2.206=264.72  pounds. 

4.  In  26.472  pounds  how  many  kilograms? 

Ans.  264.72X. 4536=120.  kilograms. 

The  meter  is  nearly  3  feet  3  inches  and  3  eighths,  or 
3  feet  3J-  inches. 

REMARKS. — The  word  Meter  means  measure,  as  in  gas- 
meter,  or  gas-measure. 

Stere  means  solid,  as  in  stereotype,  solid  type,  or  rather 
a  solid  mass  of  type.  Stereoscope,  an  instrument  to  make 
two  pictures  look  like  a  solid. 

Are  signifies  area,  a  surface  included  within  given  lines. 

Quintal  is  from  the  Latin  root  centum,  a  hundred ;  for- 
merly a  hundred  weight. 


PAGE. 

Vocabulary  of  Technical  Terms  used  in  Business 5 

Tables  of  Money,  Weights  and  Measures 15 

Notation  and  Numeration 26 

Addition...! 33 

Subtraction 43 

Multiplication 49 

Division 60 

Short  Division 61 

Percentage 68 

Long  Division 72 

Properties  of  Numbers 77 

Multiplication  by  Aliquots 79 

Easy  Fractions 82 

The  Mercantile  Profession 94 

Bills  and  Invoices 100 

Compound  Numbers 117 

Short  Methods 125 

Marking  Goods — Gain  and  Loss 132 

Commission  and  Brokerage 139 

Interest  and  forms  of  notes,  etc 142 

Simple  Interest ; 143 

Banking 154 

Checks,  Drafts,  etc 163 

Receipts 165 

Time  Tables  for  Averaging.. 168 

Discounting  Notes % 170 

True  Discount 174 

Compound  Interest 178 

Average — Equation  of  Payments ,  180 

(298) 


NELSON'S  COMMON-SCHOOL  ARITHMETIC.         299 

Accounts  Current - » 194 

Exchange 196 

Fractions. * 202 

Extinction  of  Fractions 204 

Decimals 206 

Multiplication  of  Decimals v...  210 

Addition  of  Decimals 208 

Subtraction  of  Decimals 209 

Division  of  Decimals 211 

Reduction  of  Decimals 213 

Multiplication  of  Common  Fractions 216 

Subtraction  of  Common  Fractions 222 

Division  of  Common  Fractions 219 

Addition  of  Common  Fractions... 223 

Ratio  and  Proportion 227 

Compound  Proportion 231 

Partnership 233 

Joint  Stock  Companies,  etc 235 

Importing 239 

Foreign  Exchange 240 

Insurance .-. 244 

Duodecimals 246 

Involution  and  Evolution 248 

The  Trades,  Farming,  etc 256 

.  Useful  Hints  on  Building 257 

Geometrical  Definitions 278 

Mensuration 280 

Gauging 286 

The  Metric  System, 287 


- 


-2T 


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